Bekenstein–Hawking entropy in expanding universes from black hole theorems

We show that the use of suitable theorems for black hole formation in Friedmann expanding universes leads to a modification of the Bekenstein–Hawking entropy. By adopting an argument similar to the original Bekenstein one, we write down the expression for the Bekenstein–Hawking entropy suitable for non-static isotropic expanding universes together with the equation of state of a black hole. This equation can be put in a form similar to the one of an ideal gas but with a factor depending on the Hubble radius. Moreover, we give some argument on a possible relation between our entropy expression and the Cardy–Verlinde one. Finally, we explore the possibility that primordial inflation is due to black hole evaporation in our context.