Equilibrium vs. non-equilibrium thermodynamics of the cosmological apparent horizon in modified gravity

Determine whether the cosmological apparent horizon in modified gravity theories with higher-order equations of motion, such as f(R) gravity and scalar-tensor gravity, obeys an equilibrium Clausius relation or a non-equilibrium entropy balance relation with an entropy-production term, when deriving the Friedmann equations in Friedmann–Lemaître–Robertson–Walker spacetimes.

Background

The paper compares two thermodynamic approaches to gravity beyond general relativity: the Eling–Guedens–Jacobson (EGJ) local Rindler-horizon framework and the cosmological apparent-horizon (CAH) framework in FLRW spacetimes. While both employ Clausius-type relations, their entropy-production terms have distinct origins and roles.

In modified gravity theories with higher-order equations of motion (e.g., f(R) and scalar-tensor gravity), applying the equilibrium Clausius relation on the CAH does not straightforwardly reproduce the correct Friedmann equations. Some authors introduce a non-equilibrium entropy-production term, while others redefine quantities to maintain equilibrium. This leads to ambiguity about the correct thermodynamic description at the apparent horizon.

The authors explicitly flag this ambiguity as an unresolved issue, noting that it is unclear whether the CAH in such theories should be described by equilibrium or non-equilibrium thermodynamics.