Impact of EUP correction on thermodynamics of AdS black hole

In this paper, we investigate the effect of the extended uncertainty principle (EUP) on the thermodynamic properties and stability of a charged black hole in a $4D$ Einstein–Gauss–Bonnet gravity in Anti-de Sitter ($AdS$) spacetime. We investigate the thermodynamic quantities (Mass, Hawking temperature, and entropy) associated with a quantum-corrected black hole. We compute the heat capacity and the Gibbs free energy, expressed in terms of horizon radii, to examine the local and global thermodynamic stability of black holes. The heat capacity, denoted by $C_{+}$, diverges twice at a critical radius $r=r_c$ when the temperature reaches a maximum (minimum). For $r_{+}<r_c$, $C_{+}>0$ and $G_{+}<0$, enabling the smaller black hole to attain local stability (global stability). A subsequent Hawking-Page transition is observed between the smaller black hole and thermal $AdS$ space. The EUP correction significantly impacts larger black holes but has little effect on smaller ones. Furthermore, we analyze the thermodynamics of the EUP-corrected black hole modified in an extended phase space, where the cosmological constant is interpreted as the thermodynamic pressure $(\Lambda =-8\pi P)$. We calculated the critical values of the thermodynamic variables for fixed values of $\alpha$ and $\beta$ and drawn the $P-v$ diagram. Furthermore, we examine first and second-order phase transitions by analyzing the behavior of Gibbs’s free energy–temperature plot. Below the critical point (critical pressure), a first-order phase transition is observable, concluding at the critical point, where the phase transition becomes second-order. The equation of state and critical points for black holes are analyzed, revealing similarities to those of the Van der Waals fluid.