Thermodynamic Topology of Quantum RN Black Holes

This paper presents a comprehensive exploration of the thermodynamics of black holes, focusing on foundational concepts such as free energy, entropy, and topological numbers, alongside a detailed examination of quantum RN black holes. By extending the discussion to encompass the symmetry groups SO(2), SO(3)/SO(2), and SO(3) within the framework of (f(R)) gravity, the paper offers a nuanced understanding of black hole physics. Key insights include the pivotal role of free energy and entropy in understanding the thermodynamic properties of black holes, the significance of topological numbers in determining thermodynamic stability and phase transitions, and the implications of quantum mechanics and (f(R)) gravity on traditional thermodynamic concepts. This exploration not only enriches our theoretical knowledge of black holes but also sets the stage for future empirical investigations, marking a pivotal contribution to our ongoing quest to decipher the universe's mysteries.