Emergence of spaces and the dynamic equations of FRW universes in the $f(R)$ theory and deformed Hořava-Lifshitz theory

Abstract: It has been shown that Friedmann equation of FRW universe can be derived from the idea which says cosmic space is emergent as cosmic time progresses and our universe is expanding towards the state with the holographic equipartition by Padmanabhan. In this note, we give a general relationship between the horizon entropy and the number of the degrees of freedom on the surface, we also obtain the corresponding dynamic equations by using the idea of emergence of space in the $f(R)$ theory and deformed Ho\v{r}ava-Lifshitz (HL) theory.