Friedmann equations of FRW universe in scalar-tensor gravity, f(R) gravity and first law of thermodynamics

In the paper, [R.G. Cai, S.P. Kim, JHEP 0502 (2005) 050, hep-th/0501055], it is shown that by applying the first law of thermodynamics to the apparent horizon of an FRW universe and assuming the geometric entropy given by a quarter of the apparent horizon area, one can derive the Friedmann equations describing the dynamics of the universe with any spatial curvature; using the entropy formula for the static spherically symmetric black holes in Gauss-Bonnet gravity and in more general Lovelock gravity, where the entropy is not proportional to the horizon area, one can also obtain the corresponding Friedmann equations in each gravity. In this Letter we extend the study of the above mentioned paper to the cases of scalar-tensor gravity and f(R) gravity, and discuss the implication of results.