  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.
