  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 Wienhold surface, 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 $D\ge 5$. 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 Einstein-Yang-Mills
black holes.
