  In the context of extended phase space thermodynamics and the AdS-CFT
correspondence, we consider the chemical potential ($\mu$) dual to the number
of colours ($N$) of the boundary gauge theory, in the grand canonical ensemble.
By appropriately defining $\mu$ via densities of thermodynamic quantities, we
show that it changes sign precisely at the Hawking-Page transition for
AdS-Schwarzschild and RN-AdS black holes in five dimensions, signalling the
onset of quantum effects at the transition point. Such behaviour is absent for
non-rotating black holes in four dimensions. For Kerr-AdS black holes in four
and five dimensions, our analysis points to the fact that $\mu$ can change sign
in the stable black hole region, i.e above the Hawking-Page transition
temperature, for a range of angular frequencies. We also analyse AdS black
holes in five dimensional Gauss-Bonnet gravity, and find similar features for
$\mu$ as in the Kerr-AdS case.
