Equilibrium description of thermodynamics in modified gravitational theories

We demonstrate that it is possible to obtain a picture of equilibrium thermodynamics on the apparent horizon in the expanding cosmological background for a wide class of modified gravity theories with the Lagrangian density f(R, φ,X), where R is the Ricci scalar and X = -(∇ φ)²/2 is the kinetic energy of a scalar field φ. In this framework, the horizon entropy S corresponding to equilibrium thermodynamics is proportional to the horizon area A with a constant coefficient, as in the Einstein gravity. We show that for a flat cosmological background with a decreasing Hubble parameter, S globally increases with time.