Clausius relation and Friedmann equation in FRW universe model

Abstract: It has been shown that Friedmann equation of FRW universe can be derived from the first law of thermodynamics in Einstein gravity, Gauss-Bonnet gravity, Lovelock gravity, scalar-tensor gravity and $f(R)$ gravity. Moreover, it was pointed out that the temperature of the apparent horizon can be obtained using the tunneling formalism for the corresponding observers defined by Kodama vector. In this article, we find that the energy flux through the apparent horizon can be determined by using the Kodama vector. This implies the fact that the Clausius relation and the first law of thermodynamics associated with the apparent horizon in FRW universe is relative to the Kodama observers.We illustrate the derivation of Friedmann equation, and also extend the study to the cases of Ho\v{r}ava-Lifshitz gravity and IR modified Ho\v{r}ava-Lifshitz gravity.