  In this paper, we delve into the study of thermodynamics and phase transition
of charged Gauss-Bonnet black holes within the context of anti-de Sitter (AdS)
space, with particular emphasis on the central charge's role within the dual
conformal field theory (CFT). We employ a holographic methodology that
interprets the cosmological constant and the Newton constant as thermodynamic
variables, leading to the derivation of a modified first law of thermodynamics
that incorporates the thermodynamic volume and pressure. Our findings reveal
that the central charge of the CFT is intrinsically linked to the variation of
these constants, and its stability can be ensured by simultaneous adjustment of
these constants. We further explore the phase structures of the black holes,
utilizing the free energy. Our research uncovers the existence of a critical
value of the central charge, beyond which the phase diagram displays a
first-order phase transition between small and large black holes. We also delve
into the implications of our findings on the complexity of the CFT. Our
conclusions underscore the significant role of the central charge in the
holographic thermodynamics and phase transition of charged Gauss-Bonnet black
holes. Furthermore, we conclude that while the central charge considered
provides suitable and satisfactory solutions for this black hole in 4 and 5
dimensions, it becomes necessary to introduce a unique central charge for this
structure of modified gravity. In essence, the central charge in holographic
thermodynamics is not a universal value and requires modification in accordance
with different modified gravities. Consequently, the physics of the problem
will significantly deviate from the one discussed in this article, indicating a
rich and complex landscape for future work.
