  Motivated by the recent progress on the holographic dual of the extended
thermodynamics for black holes in anti-de-Sitter (AdS) space, we investiggate
the hologrphic 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 $\mu$ conjugating to the central charge $C$, we obtain other pairs of
thermodynamics for the CFT, such as the temperature $\tilde{T}$ and the entropy
$S$, the Gauss-Bonnet coupling constant $\tilde{\alpha}$ and its conjugate
variable $ \tilde{\mathcal{A}}$, the pressure $\mathcal{P}$ and its conjugate
volume $\mathcal{V}$. In the fixed $C$, $\mathcal{V}$ and $\tilde{\alpha}$
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 $\tilde{T}-S$
criticality of the CFT are the same as those of the five-dimensional
Gauss-Bonnet AdS black hole.
