  In this paper, we have studied the thermodynamics of Gauss-Bonnet black holes
in D-dimensional $AdS$ spacetime. Here, the cosmological constant ($\Lambda$),
Newton's gravitational constant ($G$) and the Gauss-Bonnet parameter ($\alpha$)
are varied in the bulk, and a mixed first law is rewritten considering central
charge ($C$) (of dual boundary conformal theory) and its conjugate variable
utilising the gauge-gravity duality. A novel universal nature of central charge
near the critical point of black hole phase transition in Einstein's gravity
has been observed in \cite{mann1}. We observe that this universal nature breaks
when such phase transition is considered for black holes in the Gauss-Bonnet
gravity. Apart from this, treating the Gauss-Bonnet parameter as a
thermodynamic variable as suggested in \cite{kastori} in light of the
consistency between first law and the Smarr relation leads to modified
thermodynamic volume (conjugate to variable cosmological constant), adding to a
new understanding of the Van der Waals gas like behaviour of the black holes in
higher dimensional and higher curvature gravity theories. Our analysis
considers a general $D$ dimensional background. We have then imposed a greater
focus in the analysis of the phase structure of the five dimensional
Gauss-Bonnet spacetime. Our analysis also shows that the general universal
nature of the critical value of the central charge (which was present in four
dimensional $AdS$ spacetime), breaks down in case of five dimensional $AdS$
spacetime even in the absence of Gauss-Bonnet gravity. This finding indicates
the universal nature of the central charge may be a special feature of the four
dimensional $AdS$ spacetime only.
