Thermodynamics of Fractal Universe

We investigate the thermodynamical properties of the apparent horizon in a fractal universe. We find that one can always rewrite the Friedmann equation of the fractal universe in the form of the entropy balance relation $\delta Q=T_h d{S_h}$, where $\delta Q$ and $T_{h}$ are the energy flux and Unruh temperature seen by an accelerated observer just inside the apparent horizon. We find that the entropy $S_h$ consists two terms, the first one which obeys the usual area law and the second part which is the entropy production term due to nonequilibrium thermodynamics of fractal universe. This shows that in a fractal universe, a treatment with nonequilibrium thermodynamics of spacetime may be needed. We also study the generalized second law of thermodynamics in the framework of fractal universe. When the temperature of the apparent horizon and the matter fields inside the horizon are equal, i.e. $T=T_h$, the generalized second law of thermodynamics can be fulfilled provided the deceleration and the equation of state parameters ranges either as $-1 \leq q < 0$, $- 1 \leq w < - 1/3$ or as $q<-1$, $w<-1$ which are consistent with recent observations. We also find that for $T_h=bT$, with $b<1$, the GSL of thermodynamics can be secured in a fractal universe by suitably choosing the fractal parameter $\beta$.