From the perspective of higher dimensional black holes in general relativity, we consider the role of the real Riemannian geometry and describe the (in)stabilities for statistical fluctuations of background black holes in arbitrary SU(N) gauge theories. For a given ensemble of non-abelian gauge theory vacuum extremal black holes carrying a nonzero pair of gauge charge and cosmological constant, we study the Gaussian fluctuations over an equilibrium ensemble and determine the criteria for the well-defined, non-degenerate, curved and regular state-space and chemical surfaces. From the notion of the fluctuation theory, we show for the extremal black hole ensembles that the global stability and phase transition curves algebraically reduce to a set of finite polynomials, as the function of the charge. The present research offers gauge/ gravity perspective for the thermodynamic geometries of the background black holes. The perspective study includes distribution theory, local and global stability domains for colored (non)extremal black holes, higher form gauge theories, (non)static (non)abelian black holes in (gauged) supergravities and their embeddings in string theories and M-theory.