Universal Thermodynamic Topological Classes of BTZ Black Holes in Einstein and F(R) Gravity

Abstract

In this paper, we systematically study three classes of three-dimensional Bañados-Teitelboim-Zanelli (BTZ) black holes based on different gravitational frameworks and matter field structures: the Einstein-Maxwell BTZ, the F(R)-Maxwell BTZ, and the F(R)-phantom BTZ.Within the canonical and grand canonical ensemble frameworks, we construct the generalized free energy of these black hole systems and systematically analyze the asymptotic behavior of Hawking thermodynamics.The results indicate that both gravitational modifications and matter fields significantly influence black hole topology: the F(R) gravitational modification alters the intrinsic topological class of the black hole compared to the BTZ black hole under Einsteinian gravity, while different matter fields further modulate the topological classification results. Furthermore, by comparing the two types of ensembles, we find significant differences in black hole topological classes, and these differences are correlated with the gravitational framework and matter fields. In summary, the gravitational framework, matter fields, and ensemble selection are key factors governing the topological classification of black holes; all identified topological categories belong to the existing three-dimensional spacetime topological classification system, highlighting their universality. This work deepens our understanding of the thermodynamic-topological connection in three-dimensional BTZ black holes and provides a rigorous theoretical foundation for extending such topological classifications to modified gravity theories and different matter field backgrounds.