First law of thermodynamics for dynamical apparent horizons and the
  entropy of Friedmann universes
  Recently, we have generalized the Bekenstein-Hawking entropy formula for
black holes embedded in expanding Friedmann universes. In this letter, we begin
the study of this new formula to obtain the first law of thermodynamics for
dynamical apparent horizons. In this regard we obtain a generalized expression
for the internal energy $U$ together with a distinction between the dynamical
temperature $T_D$ of apparent horizons and the related one due to
thermodynamics formulas. Remarkable, when the expression for $U$ is applied to
the apparent horizon of the universe, we found that this internal energy is a
constant of motion. Our calculations thus show that the total energy of our
spatially flat universe including the gravitational contribution, when
calculated at the apparent horizon, is an universal constant that can be set to
zero from simple dimensional considerations. This strongly support the
holographic principle.
