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The thermodynamics of Universe in the Eddington-Born-Infeld (EBI) theory was restudied by utilizing the holographic-style gravitational equations that dominate the dynamics of the cosmical apparent horizon

Gravitational thermodynamics is quite an interesting question, which has attracted much attention. Recently, many studies have covered both the first and second laws of gravitational thermodynamics for the Friedmann-Robertson-Walker (FRW) Universe with a generic spatial curvature. The inspired work is the first law of thermodynamics for Universe by Cai and Kim [

Inspired by the gravitational thermodynamics in these gravitational theories [

In this paper, we derived the holographic-style dynamical equations and discussed the properties of the cosmical apparent horizon

This paper is organized as follows. In Section

Physically, apparent horizons constitute the observable boundary which is the largest boundary of Universe in an instant. Mathematically, apparent horizons are many hypersurfaces where the outward expansion rate

In order to calculate the apparent horizon of the cosmology, we use the FRW metric to describe the spatially homogeneous and isotropic Universe [

For the expanding Universe

The action of Eddington-Born-Infeld theory is given by [

Applying the Bigravity method [

Varying the Bigravity-type action (

The matter content of Universe is construed as the perfect fluid whose the energy-momentum tensor is

Depending on the field equation and the two metrics, we get the first Friedmann equation

Eq. (

With the help of (

Eq. (

Furthermore, from (

In general, the cosmical apparent horizon is not null surface, which is different from the event and particle horizon. The equation of the cosmical apparent horizon in comoving coordinates is [

Considering the properties of the quadratic function

(A) When

(B) When

(C) When

As we know, the present Universe is an accelerated expanding Universe that means the matter outside the cosmical apparent horizon may enter into the cosmical apparent horizon. Hence we considered that the timelike cosmical apparent horizon is reasonable and the range of the EoS parameter

Based on the holographic-style dynamical equations (

Applying the Misner-Sharp mass/energy

The unified first law of (equilibrium) thermodynamics is given by

In the EBI theory, the field equations can be rewritten into

We consider that the FRW metric

Naturally, because of the invariance of

where

It illustrates our hypothesis of replacing

Having obtained the unified first law

Eq. (

Finally, for the open system enveloped by

Comparing (

With the help of the first holographic-style dynamical equation (

In many papers [

In order to discuss

Based on the above assumptions, we can obtain

If

when

when

when

On the other hand, from the acceleration equation of the EBI Universe (

In a word, the physical effective entropy

In this paper, we obtained the gravitational dynamics in the EBI Universe. Firstly, we derived the holographic-style dynamical equations. Because of the present accelerated expanding Universe that means the outer matter can enter into the cosmical apparent horizon, we considered the timelike cosmical apparent horizon is reasonable and

Secondly, based on the holographic-style dynamical equations, we obtained two forms of the total energy differential in the (

Thirdly, we derived the total energy differential for the open system enveloped by

Finally, we discussed the properties of the effective entropy

In addition to all the above we would like to point out that the method of the theories we mentioned in our paper satisfying the equilibrium thermodynamics does not mean this method fits all possible theories. Actually, nonequilibrium thermodynamics might be also a good way in finding new gravitational theories [

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

We would like to thank the National Natural Science Foundation of China (Grant No. 11571342) for supporting us on this work.