https://doi.org/10.1140/epjc/s10052-025-15098-6
Regular Article - Theoretical Physics
Extended thermodynamical topology of black hole
1
Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Gansu Provincial Research Center for Basic Disciplines of Quantum Physics, Lanzhou University, 730000, Lanzhou, China
2
Institute of Theoretical Physics and Research Center of Gravitation, School of Physical Science and Technology, Lanzhou University, 730000, Lanzhou, China
a
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Received:
26
August
2025
Accepted:
17
November
2025
Published online:
2
December
2025
Abstract
Thermodynamical topology has emerged as a powerful framework for classifying the thermodynamical behavior of black holes. Three distinct yet complementary topological invariants have been employed to characterize black hole phases, spinodal curves, and critical points in black hole thermodynamics. In this work, we develop a unified framework that integrates these three topological approaches and introduce the concept of extended thermodynamical topology, providing a clear physical interpretation. As a first step, we apply this framework to black holes in Einstein gravity, systematically elucidating their phase structure in terms of topological invariants. We then extend our analysis to black holes in 7-dimensional Lovelock gravity, where novel thermodynamic phenomena naturally emerge from the topological perspective. Moreover, we explore the connection between critical exponents and the extended thermodynamical topology, uncovering a correspondence between the zeros of the k-th order vector field and the associated critical exponents. Our study demonstrates that extended thermodynamical topology offers a robust and fine-grained framework for analyzing and classifying black hole phase transitions.
© The Author(s) 2025
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