Thermodynamic Behavior of Friedmann Equation at Apparent Horizon of FRW Universe

It is shown that the differential form of Friedmann equation of a FRW universe can be rewritten as a universal form dE = T dS + W dV at apparent horizon, where E is the Misner-Sharp energy, V is the volume inside the apparent horizon, W = (ρ − P)/2 is the work density and ρ and P are energy density and pressure of the matter in the universe, respectively. From the thermodynamic identity one can derive that the apparent horizon has associated entropy S = A/4G and temperature T = κ/2π in Einstein general relativity, where A is the area of apparent horizon and κ is the surface gravity at apparent horizon. 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 rewritten to the universal form dE = T dS + W dV at the apparent horizon with entropy S being given by expression previously known via black hole thermodynamics.