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We can substitute H ( ρ + P/c $^{2}$) = − ˙ ρ/ 3 into the equation for ˙ H and integrate 

Which yields 

Or 

<img src='content_image/327836.jpg'>

<img src='content_image/327838.jpg'>

<img src='content_image/327837.jpg'>

<img src='content_image/327839.jpg'>

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 

<img src='content_image/327840.jpg'>

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. 

## 1. Continuity Equation from the First Law 

The continuity equation is derived assuming the simple form of the first law of Thermody- namics. 

<img src='content_image/327841.jpg'>

