13 We can substitute H ( ρ + P/c $^{2}$) = − ˙ ρ/ 3 into the equation for ˙ H and integrate Which yields Or This is simply the usual second Friedmann energy equation. We can find the other form of the acceleration equation by using this equation and the relationship ˙ H = ¨ a/a − H 2 to substitute into Eq. (A.7) to obtain We can see from this derivation that the Friedmann equations naturally arise from the first Law of Thermodynamics and the association of Entropy and Temperature to the apparent horizon (or Hubble Horizon for a flat universe). We also see there is a place for curvature (currently observed to be small) and for a constant of integration Λ. The continuity equation is derived assuming the simple form of the first law of Thermody- namics. 1. Continuity Equation from the First Law H$^{(}$ ˙ H − k a 2 = − 4 πG ( ρ + P/c $^{2}$)$^{)}$ = 4 πG 3 ˙ ρ (A.9) ∫ H ˙ H − k$^{∫}$ ˙ a a 3 = 4 πG 3 ∫ ˙ ρ (A.10) 1 2 ( H 2 + k a 2 ) = 4 πG 3 ρ + constant/ 2 (A.11) H 2 + kc 2 a 2 = 8 πG 3 ρ + constant → H 2 = 8 πG 3 ρ − kc 2 a 2 + Λ c 2 3 (A.12) ¨ a a = − 4 πG 3 ( ρ + 3 P c 2 ) + constant → ¨ a a = − 4 πG 3 ( ρ + 3 P c 2 ) + Λ c 2 3 (A.13) dE = − PdV dMc 2 = − PdV d ( V$_{0}$ρa $^{3}$) c 2 = − Pd ( V$_{0}$a $^{3}$) dρc $^{2}$a3 + daρc $^{2}$3 a 2 = − 3 a $^{2}$Pda dρ = − 3$^{(}$ ρ + P c 2 ) 1 a da dρ dt = − 3$^{(}$ ρ + P c 2 ) 1 a da dt