Thermodynamics in Modified Gravity Theories
  We demonstrate that there does exist an equilibrium description of
thermodynamics on the apparent horizon in the expanding cosmological background
for a wide class of modified gravity theories with the Lagrangian density $f(R,
\phi, X)$, where $R$ is the Ricci scalar and $X$ is the kinetic energy of a
scalar field $\phi$. This comes from a suitable definition of an energy
momentum tensor of the "dark" component obeying the local energy conservation
law in the Jordan frame. It is shown that the equilibrium description in terms
of the horizon entropy $S$ is convenient because it takes into account the
contribution of the horizon entropy $\hat{S}$ in non-equilibrium thermodynamics
as well as an entropy production term.
