Friedmann equations and thermodynamics of apparent horizons

With the help of a masslike function which has a dimension of energy and is equal to the Misner-Sharp mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon dE = T(A)dS(A) can be derived from the Friedmann equation in various theories of gravity, including the Einstein, Lovelock, nonlinear, and scalar-tensor theories. This result strongly suggests that the relationship between the first law of thermodynamics of the apparent horizon and the Friedmann equation is not just a simple coincidence, but rather a more profound physical connection.