It is well known that there is a close connection between gravity and thermodynamics. This is especially true for the physics of black holes, whose thermodynamics is more or less generally accepted. It is determined by the temperature of the Hawking radiation from the horizon of the black hole, and the corresponding entropy is proportional to the area of the horizon, $S_{\rm BH}=A/4G$. However, in the case of the thermodynamics of the cosmological horizon in an expanding de Sitter Universe, the situation is not so clear. There are several different approaches to de Sitter thermodynamics with different assumptions about its temperature and entropy. The reason is that, unlike a black hole, the de Sitter state is not a localized object. It cannot be considered as a region bounded by the cosmological horizon. The de Sitter state is an unbounded symmetric state with constant scalar curvature.

It is usually assumed that the corresponding temperature of the de Sitter state is related to the temperature of the Hawking radiation from the cosmological horizon, the Gibbons-Hawking temperature $T_{\rm GH}=H/2\pi$, where $H$ is the Hubble parameter. However, if we consider the behavior of any object, for example an atom, placed in a de Sitter medium, it turns out that this object perceives the de Sitter vacuum as a heat bath with twice the Gibbons-Hawking temperature, $T=2T_{\rm GH}=H/\pi$. Since all points in de Sitter space are equivalent both inside and outside the horizon, this temperature is uniform, being the same for all local observers. Thus, a factor of two provides the difference between two physical temperatures: the local temperature $T$ and the temperature $T_{\rm GH}$ of the Hawking radiation from the cosmological horizon. This is one of the contradictions present in the construction of the thermodynamics of the de Sitter state.

Here we discuss the connections between these two thermodynamics, local thermodynamics and the thermodynamics of the Hubble volume, the volume of the region inside the cosmological horizon. (i) The local temperature is exactly twice the Gibbons-Hawking temperature. This connection has a simple explanation, following from de Sitter symmetry. (ii) There is a holographic connection between these thermodynamics. The entropy density integrated over the Hubble volume coincides with the entropy of the horizon, $S_{\rm Hubble}=S_{\rm horizon}=A/4G$, where $A$ is the area of the cosmological horizon. (iii) There is also a connection between the first law of local thermodynamics and the first law of global thermodynamics. Due to de Sitter symmetry, the first law is valid for an arbitrary volume $V$, which can be smaller or larger than the Hubble volume. This first law can also be applied to Hubble volume. In this case, the first law is expressed in terms of the entropy of the horizon. It is important that in both cases the thermodynamics is determined by the temperature $T=H/\pi$.

This consideration was also applied to the contracting de Sitter, for which $S_{\rm Hubble}=S_{\rm horizon}=-A/4G$. The entropy of the contracting de Sitter is negative, since its horizon is similar to the horizon of a white hole.

G.E.Volovik
JETP Letters 121, Issue 10 (2025)