  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.
