Black holes thermodynamics with CFT re-scaling

In this paper, we study the thermodynamic behavior of charged AdS black holes in a conformal holographic extended thermodynamic. Our setup is constructed using a new dictionary that relates AdS black hole quantities to the corresponding dual conformal field theory (CFT) one, with the conformal factor being treated as a variable thermodynamic. In this thermodynamic study, we investigate the critical phenomena of charged AdS black holes and their relationship to the central charge value, $C$. Additionally, we examine the phase transitions and black hole stability using the free energy and the heat capacity, respectively. Furthermore, by examining the chemical potential, we establish criteria that differentiate between quantum and classical black hole behaviors. Our setup highlights one of the key findings, namely traditional black hole thermodynamics acts as a boundary between quantum and classical regimes.