Subtleties in non-equilibrium horizon thermodynamics of modified gravity theories
Thermodynamic interpretations of gravity often arise from applying the Clausius relation to spacetime horizons. In modified gravity theories with higher-order equations of motion, such as f(R) and scalar-tensor gravity, this relation generally acquires additional entropy-production term. In this context, two distinct formulations have been proposed in literature: the non-equilibrium approach of Eling, Guedens, and Jacobson based on local Rindler horizons, and the thermodynamic formulation of cosmological apparent horizons in FLRW spacetimes. In this article, we present a detailed analysis of these approaches, and show that, even though both employ identical entropy balance relations that resemble non-equilibrium thermodynamics, the exact origin and role of each entropy-production term is fundamentally different. In the Rindler-horizon framework the extra term follows directly from consistency requirements related to the Bianchi identity, whereas in the apparent-horizon approach it is introduced solely to recover the Friedmann equations. Furthermore, we will see that the latter non-equilibrium contribution enters directly into dynamical equations of gravity, while the former does not. Finally, we also highlight the fact that thermodynamic descriptions of horizons in such modified gravity are not unique, and that equilibrium, and non-equilibrium descriptions can arise from different choices of thermodynamic variables. A clear understanding of these distinctions is therefore crucial for establishing a consistent and physically meaningful thermodynamic foundation for gravity beyond general relativity.