Thermodynamics of de Sitter Black Holes: Thermal Cosmological Constant

TL;DR

This work examines the thermodynamics of black hole event horizons and cosmological horizons in asymptotically de Sitter spacetimes using Teitelboim's Euclidean method, treating the cosmological constant $\Lambda$ as a thermodynamic state variable. It derives generalized Smarr formulas for both horizons and establishes consistent first laws with horizon-specific temperatures, entropies, angular velocities, and conjugate volumes, revealing that $\Lambda$ must vary (and tends to decrease) under quantum effects to satisfy the generalized second law. The analysis shows a formal analogy with AdS thermodynamics under the substitution $l^2 \to -l^2$, suggesting possible analytic continuation between AdS and dS cases, while challenging fixed-$\Lambda$ interpretations of dS/CFT. The results imply a link between horizon thermodynamics, vacuum decay, and inflationary dynamics, and point to Hawking-Page–like phase transition behavior in de Sitter spacetimes as a subject for future study.

Abstract

We study the thermodynamic properties associated with the black hole event horizon and the cosmological horizon for black hole solutions in asymptotically de Sitter spacetimes. We examine thermodynamics of these horizons on the basis of the conserved charges according to Teitelboim's method. In particular, we have succeeded in deriving the generalized Smarr formula among thermodynamical quantities in a simple and natural way. We then show that cosmological constant must decrease when one takes into account the quantum effect. These observations have been obtained if and only if cosmological constant plays the role of a thermodynamical state variable. We also touch upon the relation between inflation of our universe and a phase transition of black holes.