Dynamics of the Cosmological Apparent Horizon: Surface Gravity & Temperature

In the context of thermodynamics applied to our cosmological apparent horizon, we explicit in greater details our previous work which established the Friedmann Equations from projection of Hayward's Unified First Law. In particular, we show that the dynamical Hayward-Kodama surface gravity is perfectly well-defined and is suitable for this derivation. We then relate this surface gravity to a physical notion of temperature, and show this has constant, positive sign for any kind of past-inner trapping horizons. Hopefully this will clarify the choice of temperature in a dynamical Friedmann-Lema\^itre-Roberston-Walker spacetime.