Holographic CFT phase transitions and criticality for charged Gauss–Bonnet AdS black holes in the ensemble at fixed $$(C, \mathcal {V}, \tilde{Q}, \tilde{\mathcal {A}})$$

Abstract We study the holographic dual of the extended thermodynamics of spherically symmetric, charged Gauss–Bonnet AdS black holes in the context of the AdS/CFT correspondence. Compared to Einstein’s theory of gravity, Gauss–Bonnet gravity introduces higher-order curvature terms. The coupling constants of these higher-order curvature terms $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> can serve as new thermodynamic quantities, which will also be dual to thermodynamic quantities on the boundary CFT, a feature not present in the CFT dual to Einstein’s gravity previously. Based on the holographic dictionary, we studied the critical behavior and phase transition of the CFT description of the charged Gauss–Bonnet black holes in $$d=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and $$d=5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> in the ensemble at fixed $$(C, \mathcal {V}, \tilde{Q}, \tilde{\mathcal {A}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The interesting behaviour of free energy stems from the fact that the constraints we introduced to handle the gravitational constant on CFT and the AdS radius differ from conventional approaches. Using the criticality equation, we numerically found the critical points of the zeroth-order and first-order phase transition for $$\tilde{\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> . The relationships between conjugate thermodynamic pairs (equation of state) were also examined. In the case of the $$p-\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> , $$\tilde{T}-\tilde{S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>-</mml:mo> <mml:mover> <mml:mi>S</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> </mml:math> and $$\tilde{\varPhi }-\tilde{Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>Φ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>-</mml:mo> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> </mml:math> conjugate pairs, characteristics that are analogous to the first-order phase transition of van der Waals fluids were found.