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

TL;DR

The paper analyzes de Sitter thermodynamics across general spacetime dimensions, introducing a local temperature $T=H/\pi$ to define the entropy of the Hubble volume. It shows that in $(3+1)$ dimensions the Gibbons–Hawking entropy $S_{GH}=A/(4G)$ is recovered, but for arbitrary $d+1$ dimensions the on-shell entropy of the Hubble volume is $S_H(d)=(d-1)A/(8G)$, necessitating a modified horizon entropy to preserve holographic relations. The authors connect this modification to the first law of de Sitter thermodynamics, consider contracting de Sitter with negative entropy, explore connections to Wald entropy, and discuss horizonless gravastar models with de Sitter cores under non-extensive Tsallis–Cirto statistics. They also examine thermodynamic and quantum fluctuations of the horizon area, identifying a dimension-dependent canonical pair and highlighting implications for horizon thermodynamics and cosmological models. Overall, the work suggests a dimension-dependent revision of horizon entropy and emphasizes the deep links between local thermodynamics, holography, and horizon fluctuations in de Sitter space.

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 $T=H/π$. 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 $T_{\rm GH}=H/2π$. 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 $d=3$.