We investigate how deviations from the Bekenstein–Hawking entropy modify black-hole spacetimes through the recently proposed entropy-geometry correspondence. For four representative modified entropies, namely Barrow, Rényi, Kaniadakis, and logarithmic, we derive the corresponding effective metrics and analyze their thermodynamic and topological classification using the off-shell free energy and winding numbers. We show that Barrow and Rényi entropies yield a single unstable sector with global charge $$W=-1$$ , while logarithmic and Kaniadakis corrections produce canceling defects with $$W=0$$ , revealing topological structures absent in the Schwarzschild case. Using the modified metrics, we further calculate the photon-sphere radius and shadow size, showing that each modified entropy relation induces characteristic optical shifts. Thus, by comparing with Event Horizon Telescope observations of Sgr A $$^*$$ , we extract new bounds on all entropy-deformation parameters. Our results demonstrate that thermodynamic topology, together with photon-sphere phenomenology, offers a viable way to test generalized entropy frameworks and probe departures from the Bekenstein-Hawking area law.

India

Ankit Anand

North Macedonia

Kimet Jusufi

Greece

Cyprus

Spyros Basilakos

Chile

China

Emmanuel Saridakis

Topological and optical signatures of modified black-hole entropies

In this work we apply the gravity-thermodynamics approach for the case of generalized mass-to-horizon entropy, which is a two-parameter extension of Bekenstein–Hawking entropy that arises from the extended mass-to-horizon relation, that is in turn required in order to have consistency with the Clausius relation. We extract the modified Friedmann equations and we obtain an effective dark energy sector arising from the novel terms. We derive analytical solutions for the dark energy density parameter, the dark energy equation-of-state parameter, and the deceleration parameter, and we show that the Universe exhibits the usual thermal history with the succession of matter and dark energy epochs. Additionally, depending on the value of the entropy parameters, the dark energy equation-of-state parameter can either lie in the phantom regime at high redshifts entering into the quintessence regime at small redshifts, or it can lie in the quintessence regime at high redshifts and experience the phantom-divide crossing at small redshifts, while in the far future in all cases it asymptotically obtains the cosmological constant value $$-1$$ . Finally, we perform observational confrontation with Supernova Type Ia (SNIa), Cosmic Chronometers (CC) and Baryonic Acoustic Oscillations (BAO) datasets, showing that the scenario is in agreement with observations.

Andreas Lymperis

Maria Petronikolou

Modified cosmology through spacetime thermodynamics and generalized mass-to-horizon entropy

$\textit{f}$($\textit{Q}$) and $\textit{f}$($\textit{T}$) gravity are based on fundamentally different geometric frameworks, yet they exhibit many similar properties. This article provides a comprehensive summary and comparative analysis of the various theoretical branches of torsional gravity and non-metric gravity, which arise from different choices of affine connection. We identify two types of background-dependent and classical correspondences between these two theories of gravity. The first correspondence is established through their equivalence within the Minkowski spacetime background. To achieve this, we develop the tetrad-spin formulation of $\textit{f}$($\textit{Q}$) gravity and derive the corresponding expression for the spin connection. The second correspondence is based on the equivalence of their equations of motion. Utilizing a metric-affine approach, we derive the general affine connection for static and spherically symmetric spacetime in $\textit{f}$($\textit{Q}$) gravity and compare its equations of motion with those of $\textit{f}$($\textit{T}$) gravity. Among others, our results reveal that, $\textit{f}$($\textit{T}$) solutions are not simply a subset of $\textit{f}$($\textit{Q}$) solutions; rather, they encompass a complex solution beyond $\textit{f}$($\textit{Q}$) gravity in black hole background.

Cheng Wu

Japan

Xin Ren

Yuhang Yang

Yu-Min Hu

Background-dependent and classical correspondences between $\textit{f}$($\textit{Q}$) and $\textit{f}$($\textit{T}$) gravity

We use Weyl connection and Weyl geometry in order to construct novel modified gravitational theories. In the simplest case where one uses only the Weyl-connection Ricci scalar as a Lagrangian, the theory recovers general relativity. However, by upgrading the Weyl field to a dynamical field with a general potential and/or general couplings constructed from its trace, leads to new modified gravity theories, where the extra degrees of freedom arise from the Weyl field. Additionally, since the Weyl-connection Ricci scalar differs from the Levi–Civita Ricci scalar by terms up to first derivatives of the Weyl field, the resulting field equations for both the metric and the Weyl field are of second order, and thus the theory is free from Ostrogradsky ghosts. Finally, we construct the most general theory, namely the $$f(\tilde{R},\mathcal {A})$$ gravity, which is also ghost free. Applying the above classes of theories at a cosmological framework we obtain an effective dark energy sector of geometrical origin. In the simplest class of theories we are able to obtain an effective cosmological constant, and thus we recover $$\Lambda $$ CDM paradigm, nevertheless in more general cases we acquire a dynamical dark energy. These theories can reproduce the thermal history of the Universe, and the corresponding dark energy equation-of-state parameter presents a rich behavior.

Gerasimos Kouniatalis

Modified gravity from Weyl connection and the $$f(R,\mathcal {A})$$ extension

We investigate entanglement islands and the Page curve in the framework of Horndeski gravity on a Karch-Randall braneworld background. In particular, treating the holographic boundary conformal field theory analytically we find that the Horndeski parameters significantly alter the behavior of the Page curve compared to standard general relativity, a feature caused by the nontrivial geometry induced by the Horndeski scalar field. Interestingly enough, the geometry far from the AdS limit plays a more significant role compared to previous studies. This suggests that Horndeski gravity introduces important modifications to the distribution of quantum information in the holographic model. Finally, we claim that holographic consistency can be used reversely to impose constraints on Horndeski gravity itself, providing a new tool for probing the validity of modified gravity theories.

Brazil

Fabiano Santos

Iran

Azerbaijan

Behnam Pourhassan

Ukraine

Oleksii Sokoliuk

Alexander Baransky

Türkiye

Emre Kahya

Holographic boundary conformal field theory within Horndeski gravity

We present for the first time a Friedmann-like construction in the framework of an osculating Finsler–Randers–Sasaki (F–R–S) geometry. In particular, we consider a vector field in the metric on a Lorentz tangent bundle, and thus the curvatures of horizontal and vertical spaces, as well as the extra contributions of torsion and non-linear connection, provide an intrinsic richer geometrical structure, with additional degrees of freedom, that lead to extra terms in the field equations. Applying these modified field equations at a cosmological setup we extract the generalized Friedmann equations for the horizontal and vertical space, showing that we obtain an effective dark energy sector arising from the richer underlying structure of the tangent bundle. Additionally, as it is common in Finsler-like constructions, we obtain an effective interaction between matter and geometry. Finally, we consider a specific model and we show that it can describe the sequence of matter and dark-energy epochs, and that the dark-energy equation of state can lie in the quintessence or phantom regimes, or cross the phantom divide.

E. Kapsabelis

P. Stavrinos

Finsler–Randers–Sasaki gravity and cosmology

We present how Tsallis cosmology can alleviate both  $$H_0$$  <math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math>   and  $$\sigma _8$$  <math> <msub> <mi>σ</mi> <mn>8</mn> </msub> </math>   tensions simultaneously. Such a modified cosmological scenario is obtained by the application of the gravity-thermodynamics conjecture, but using the non-additive Tsallis entropy, instead of the standard Bekenstein–Hawking one. Hence, one obtains modified Friedmann equations, with extra terms that depend on the new Tsallis exponent  $$\delta $$  <math> <mi>δ</mi> </math>   that quantifies the departure from standard entropy. We show that for particular  $$\delta $$  <math> <mi>δ</mi> </math>   choices we can obtain a phantom effective dark energy, which is known to be one of the sufficient mechanisms that can alleviate  $$H_0$$  <math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math>   tension. Additionally, for the same parameter choice we obtain an increased friction term and an effective Newton’s constant smaller than the usual one, and thus the  $$\sigma _8$$  <math> <msub> <mi>σ</mi> <mn>8</mn> </msub> </math>   tension is also solved. These features act as a significant advantage of Tsallis modified cosmology.

Alleviating both  $$H_0$$  <math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math>   and  $$\sigma _8$$  <math> <msub> <mi>σ</mi> <mn>8</mn> </msub> </math>   tensions in Tsallis cosmology

We investigate the cosmological implications of f(Q) gravity, which is a modified theory of gravity based on non-metricity, in non-flat geometry. We perform a detailed dynamical-system analysis keeping the f(Q) function completely arbitrary. As we show, the cosmological scenario admits a dark-matter dominated point, as well as a dark-energy dominated de Sitter solution which can attract the Universe at late times. However, the main result of the present work is that there are additional critical points which exist solely due to curvature. In particular, we find that there are curvature-dominated accelerating points which are unstable and thus can describe the inflationary epoch. Additionally, there is a point in which the dark-matter and dark-energy density parameters are both between zero and one, and thus it can alleviate the coincidence problem. Finally, there is a saddle point which is completely dominated by curvature. In order to provide a specific example, we apply our general analysis to the power-law case, showing that we can obtain the thermal history of the Universe, in which the curvature density parameter may exhibit a peak at intermediate times. These features, alongside possible indications that non-zero curvature could alleviate the cosmological tensions, may serve as advantages for f(Q) gravity in non-flat geometry.

Hamid Shabani

Malaysia

Avik De

Tee-How Loo

Cosmology of f(Q) gravity in non-flat Universe

We investigate sufficient conditions under which cubic gravity is healthy and viable at the perturbation level. We perform a detailed analysis of the scalar and tensor perturbations. We impose the requirement that the two scalar potentials, whose ratio is the post-Newtonian parameter  $$\gamma $$  <math> <mi>γ</mi> </math>  , should deviate only minimally form general relativity. Additionally, concerning tensor perturbations we impose satisfaction of the LIGO-VIRGO and Fermi Gamma-ray Burst observations, and thus we result to a gravitational-wave equation with gravitational-wave speed equal to the speed of light, and where the only deviation from general relativity appears in the dispersion relation. Furthermore, we show that cubic gravity exhibits an effective Newton’s constant that depends on the model parameter, on the background evolution, and on the wavenumber scale. Hence, by requiring its deviation from the standard Newton’s constant to be within observational bounds we extract the constraints on the single coupling parameter  $$\beta $$  <math> <mi>β</mi> </math>  .

Petros Asimakis

Building cubic gravity with healthy and viable scalar and tensor perturbations

We investigate the effect of the dark matter (DM) halos collisions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on the stochastic gravitational wave background. We first calculate the gravitational wave signal of a single collision event, assuming point masses and linear perturbation theory. Then we proceed to the calculation of the energy spectrum of the collective effect of all dark matter collisions in the Universe. Concerning the DM halo collision rate, we show that it is given by the product of the number density of DM halos, which is calculated by the extended Press–Schechter (EPS) theory, with the collision rate of a single DM halo, which is given by simulation results, with a function of the linear growth rate of matter density through cosmological evolution. Hence, integrating over all mass and distance ranges, we finally extract the spectrum of the stochastic gravitational wave background created by DM halos collisions. As we show, the resulting contribution to the stochastic gravitational wave background is of the order of  $$h_{c} \approx 10^{-29}$$  <math> <mrow> <msub> <mi>h</mi> <mi>c</mi> </msub> <mo>≈</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>29</mn> </mrow> </msup> </mrow> </math>   in the band of  $$f \approx 10^{-15} Hz$$  <math> <mrow> <mi>f</mi> <mo>≈</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>15</mn> </mrow> </msup> <mi>H</mi> <mi>z</mi> </mrow> </math>  . However, in very low frequency band, it is larger. With current observational sensitivity it cannot be detected.

Qiming Yan

Yaqi Zhao

Stochastic gravitational wave background from the collisions of dark matter halos