Entropy of de Sitter (d+1)-spacetime: modification of the Gibbons-Hawking entropy of cosmological horizon

Abstract: We discuss the connection between the entropy of the Hubble volume in de Sitter spacetime and the Gibbons-Hawking entropy $S_{\rm GH}=A/4G$ associated with the cosmological horizon. In (3+1) spacetime, these two entropies coincide, and hence the Gibbons-Hawking conjecture holds. This provides physical meaning and a natural explanation to the Gibbons-Hawking entropy -- it is the entropy in the volume $V_H$ bounded by the cosmological horizon. Here we consider whether the Gibbons-Hawking conjecture remains valid for the de Sitter state in general $d+1$ spacetime. To do this, we use local de Sitter thermodynamics, characterized by a local temperature $T=H/\pi$. This temperature is not related to the horizon: it is the temperature of local activation processes, such as the ionization of an atom in the de Sitter environment, which occur deep within the cosmological horizon. This local temperature is twice the Gibbons-Hawking temperature $T_{\rm GH}=H/2\pi$. Two different ways of calculations of the entropy of the Hubble volume were considered: integration of the local entropy density over the Hubble volume and the first law of the de Sitter thermodynamics. In both cases we found that the entropy of the Hubble volume is $S_H=(d-1)A/8G$, which modifies the Gibbons-Hawking entropy of horizon. The original form of the Gibbons-Hawking is valid only for $d=3$.