Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe

It is shown that the differential form of Friedmann equation of a FRW universe can be rewritten as the first law of thermodynamics $dE=TdS+WdV$ at apparent horizon, where $E=ensuremath{rho}V$ is the total energy of matter inside the apparent horizon, $V$ is the volume inside the apparent horizon, $W=(ensuremath{rho}ensuremath{-}P)/2$ is the work density, $ensuremath{rho}$ and $P$ are energy density and pressure of matter in the universe, respectively. From the thermodynamic identity one can derive that the apparent horizon ${stackrel{texttildelow{}}{r}}_{A}$ has associated entropy $S=A/4G$ and temperature $T=ensuremath{kappa}/2ensuremath{pi}$ in Einstein general relativity, where $A$ is the area of apparent horizon and $ensuremath{kappa}$ is the surface gravity at apparent horizon of FRW universe. We extend our procedure to the Gauss-Bonnet gravity and more general Lovelock gravity and show that the differential form of Friedmann equations in these gravities can also be written as $dE=TdS+WdV$ at the apparent horizon of FRW universe with entropy $S$ being given by expression previously known via black hole thermodynamics.