Both the Barrow and Tsallis  $$\delta $$  <math> <mi>δ</mi> </math>   entropies are one-parameter generalizations of the black-hole entropy, with the same microcanonical functional form. The ensuing deformation is quantified by a dimensionless parameter  $$\Delta $$  <math> <mi>Δ</mi> </math>  , which in the case of Barrow entropy represents the anomalous dimension caused by quantum fluctuations over the horizon surface, while in Tsallis’ case, it describes the deviation of the holographic scaling from extensivity. Here, we utilize the gravity-thermodynamics conjecture with the Barrow–Tsallis entropy to investigate the implications of the related modified Friedmann equations on the spectrum of primordial gravitational waves. We show that, with the experimental sensitivity of the next generation of gravitational wave detectors, such as the Big Bang Observer, it will be possible to discriminate deviations from the  $$\Lambda $$  <math> <mi>Λ</mi> </math>  CDM model up to  $$\Delta \lesssim \mathcal {O}(10^{-3})$$  <math> <mrow> <mi>Δ</mi> <mo>≲</mo> <mi>O</mi> <mo>(</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo>)</mo> </mrow> </math>  . In this limit, Barrow–Tsallis entropy reduces to a logarithmic correction to holographic scaling, which is nearly universally predicted by both entanglement entropy calculations in the UV regime and by several candidate theories of quantum gravity. Hence, our considerations and results are expected to have general validity in the quantum gravity framework.