Holographic thermodynamics of a five-dimensional neutral Gauss–Bonnet AdS black hole

Abstract Motivated by the recent progress on the holographic dual of the extended thermodynamics for black holes in anti-de Sitter (AdS) space, we investigate the holographic thermodynamics for the five-dimensional neutral Gauss–Bonnet AdS black hole in the context of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. Through the extended bulk thermodynamics for the five-dimensional Gauss–Bonnet AdS black hole, we derive the first law of the CFT thermodynamics which is obtained by directly translating the arbitrary conformal factors in the dual CFT. In addition to the newly defined chemical potential $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> conjugating to the central charge C , we obtain other pairs of thermodynamics for the CFT, such as the temperature $${\tilde{T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> and the entropy S , the Gauss–Bonnet coupling constant $${\tilde{\alpha }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> and its conjugate variable $$\tilde{\mathcal {A}}$$ <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 pressure $$\mathcal {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> and its conjugate volume $$\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . In the fixed C , $$\mathcal {V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> and $${\tilde{\alpha }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>α</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> canonical ensemble, we obtain the canonical description of the CFT thermodynamics and observe the standard swallowtail behavior in the Helmholtz free energy vs the temperature plot. The self-intersection point of the Helmholtz free energy indicates the phase transition between the high and low entropy states of the CFT. By using Maxwell’s equal area law, we get the critical point and coexistence curve for the high and low entropy phases of the CFT. Besides, we get the critical exponents for the CFT, and find that the critical point and critical exponents associated with the $${\tilde{T}}-S$$ <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:mi>S</mml:mi> </mml:mrow> </mml:math> criticality of the CFT are the same as those of the five-dimensional Gauss–Bonnet AdS black hole.