Applicability of holography in thermodynamic equilibrium

For a strongly coupled system that has gravity dual description, we show the standard holographic dictionary yields nonnegative susceptibility when is in thermodynamic equilibrium and correlation function absolutely integrable. When no spontaneous condensation or $\mathbb{Z}_2$-symmetry breaking, find ``trace energy condition" violated many cases. There normalized grand potential density monotonic as accessing to lower scales, providing candidate $c$-function characterizing number of effective degrees freedom. Finally, discuss ``paradox'' raising by negative holography its resolution.