Thermodynamics of Flat 4D Einstein-Gauss-Bonnet Black Hole with R\'enyi Entropy: An RPST-like formalism

TL;DR

This paper explores the thermodynamics of flat 4D Einstein-Gauss-Bonnet black holes using Rényi entropy, revealing Van der Waals-like phase transitions and a formal analogy to AdS black hole thermodynamics within a novel RPST-like framework.

Contribution

It introduces a RPST-like formalism for flat black holes with Rényi entropy, uncovering thermodynamic dualities and phase behaviors similar to AdS black holes, bridging flat-space and holographic thermodynamics.

Findings

Discovery of Van der Waals-like phase transitions in flat black holes.

Identification of a thermodynamic duality between Rényi parameter and response potential.

Structural similarity between flat and AdS black holes in thermodynamic topology.

Abstract

We investigate the thermodynamics of asymptotically flat black holes in four-dimensional Einstein-Gauss-Bonnet (4D-EGB) gravity using R\'enyi entropy as a non-extensive generalization of the Bekenstein-Hawking entropy. The resulting thermodynamic structure, formulated within a restricted phase space-like (RPST-like) framework, reveals a striking resemblance to the thermodynamics of AdS black holes in the standard RPST formalism. In particular, we identify a thermodynamic duality between the R\'enyi deformation parameter $\beta$ and a conjugate response potential $\zeta$, analogous to the central charge and chemical potential in holographic theories. An extensive thermodynamic analysis in both fixed charge-$(\tilde{Q})$ and fixed potential-$(\tilde{\Phi})$ ensembles reveal Van der Waals-like first-order phase transitions which is an unexpected feature for asymptotically flat black holes. Furthermore, through the formalism of geometrothermodynamics (GTD) and thermodynamic topology, It is shown that the R\'enyi modified flat black hole mimics, in both its thermodynamic topology and geometry, the features of its counterparts in the 4D-EGB AdS black hole under RPST, reinforcing the structural similarity between these seemingly different systems. Our findings point to a deeper correspondence between non-extensive entropy and holographic thermodynamics, suggesting that R\'enyi entropy may serve as a natural bridge between flat-space black hole thermodynamics and AdS holography.