Thermodynamics, Phase Transition and Joule Thomson Expansion of 4-D Gauss-Bonnet AdS Black Hole

We explore the thermodynamic and phase transition properties of asymptotically AdS black holes within Einstein-Gauss-Bonnet gravity, focusing on Joule Thomson expansion. Thermodynamics is studied in the extended phase space, where the cosmological constant serves as thermodynamic pressure. We observe that the black hole undergoes a phase transition similar to that of a van der Waals system. We analyze charged and neutral cases separately to distinguish the effect of charge and Gauss Bonnet parameter on critical behavior and examine the phase structure. We find that the Gauss-Bonnet coupling parameter behaves similarly to black hole charge or spin, guiding the phase structure. To understand the underlying phase structure determined by the Gauss-Bonnet coefficient $\alpha$, we introduce a new order parameter. We discover that the change in the conjugate variable to the Gauss-Bonnet parameter acts as an order parameter, demonstrating a critical exponent of $1/2$ in the vicinity of the critical point. Since the phase structure is analogous to that of a van der Waals fluid, we investigate the Joule-Thomson expansion of the black hole. We analytically study the Joule-Thomson expansion, focusing on three key characteristics: the Joule-Thomson coefficient, inversion curves, and isenthalpic curves. We obtain isenthalpic curves in the $T-P$ plane and illustrate the cooling-heating regions.