We study thermodynamic state-space geometry of the black holes in string theory and M-theory. For a large number of microstates, we analyze the intrinsic state-space geometry for (i) extremal and non-extremal black branes in string theory, (ii) multi-centered black brane configurations, (iv) small black holes with fractional branes, and (v) fuzzy rings in the setup of Mathur’s fuzzballs and subensemble theory. We extend our analysis for the black brane foams and bubbling black brane solutions in M-theory. We discuss the nature of state-space correlations of various black brane configurations, and show that the notion of state-space manifolds describes the associated coarse-grained interactions of the corresponding microscopic CFT data.

From the perspective of the statistical fluctuation theory, we explore the role of the thermodynamic geometries and vacuum (in)stability properties for the topological Einstein–Yang–Mills black holes. In this paper, from the perspective of the state-space surface and chemical Weinhold surface of higher dimensional gravity, we provide the criteria for the local and global statistical stability of an ensemble of topological Einstein–Yang–Mills black holes in arbitrary spacetime dimensions D ≥ 5. Finally, as per the formulations of the thermodynamic geometry, we offer a parametric account of the statistical consequences in both the local and global fluctuation regimes of the topological extremal Einstein–Yang–Mills black holes.