Thermodynamic behavior of field equations for f(R) gravity

Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics at apparent horizon of Friedmann-Robertson-Walker (FRW) universe, which indicates intrinsic thermodynamic properties of apparent horizon of spacetime. In the present Letter we deal with the so-called f(R) gravity, whose action is a function of the curvature scalar R. In the setup of FRW universe, we show that the field equations can also be cast to a similar form, dE=TdS+WdV+TdS¯, at the apparent horizon, where W=(ρ−P)/2, E is the energy of perfect fluid with energy density ρ and pressure P inside the apparent horizon. T and S=Af′(R)/4G are temperature and entropy associated with the apparent horizon, respectively. Compared to the case of Einstein's general relativity, an additional term dS¯ appears here. The appearance of the additional term is consistent with the argument recently given by Eling et al. [C. Eling, R. Guedens, T. Jacobson, Phys. Rev. Lett. 96 (2006) 121301, gr-qc/0602001] that the horizon thermodynamics is non-equilibrium one for the f(R) gravity.