Thermodynamic Topology and Photon Spheres Analysis of Black Holes in Brane-World: Insights from Barrow Entropy

TL;DR

This paper investigates the thermodynamics and topology of brane-world black holes with Barrow entropy, revealing how entropy deformation affects phase transitions, heat capacity, and topological charges, especially in de Sitter models.

Contribution

It introduces a detailed analysis of black hole thermodynamics and topology in brane-world scenarios considering Barrow entropy, highlighting the impact of fractal horizon geometry and entropy deformation.

Findings

Divergence points in heat capacity depend on the type of entropy used.

Topological charge is predominantly -1, influenced mainly by dark matter parameters.

Cosmological horizons in dS models prevent stable photon spheres and affect topological classifications.

Abstract

We explore the thermodynamics and geothermodynamics of black holes with Barrow entropy in a brane-world scenario, where the horizon geometry of the black hole is regarded as a fractal structure. Our analysis reveals the behavior of heat capacity, identifying both bound and divergence points. For the Bekenstein-Hawking entropy, the divergence point exhibits smooth behavior, indicating no phase transition. In contrast, we observe divergence with Barrow entropy as the deformation parameter increases, confirming the presence of a zero point in heat capacity through various thermodynamic geometry formalisms. Additionally, we delve into thermodynamic topology, detailing the classification of black holes in the brane-world context and comparing their characteristics determined from the Bekenstein-Hawking and the Barrow entropy. Notably, fixing the deformation and cosmological parameters results in a topological charge $-1$ predominately by the dark matter parameter, which remains unaffected despite variations in other parameters. In the dS model, the cosmological horizon prevents stable photon spheres, making topological charges of $0$ and $+1$ unattainable. Incremental increases in the cosmological parameter reduce the dark matter parameter-dominated region.