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 = T dS + WdV + T dS, at the apparent horizon, where W = (rho = P)/2, E is the energy of perfect fluid with energy density p 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. (C) 2007 Elsevier B.V. All rights reserved.