  Thermodynamic Riemannian geometry provides great insights into the
microscopic structure of black holes (BHs). One such example is the Ruppeiner
geometry which is the metric space comprising the second derivatives of entropy
with respect to other extensive variables of the system. Reissner-Nordstr\"om
black holes (RNBHs) are known to be endowed with a flat Ruppeiner geometry for
all higher spacetime dimensions. However this holds true if one invokes
classical gravity where the semi-classical Bekenstein-Hawking entropy best
describes the thermodynamics of the system. If the much deeper quantum gravity
and string theories entail modifications to BH entropy, this prompts the
question whether the Ruppeiner flatness associated with higher dimensional
RNBHs still persists. We investigate this problem by considering
non-perturbative (exponential) and perturbative (logarithmic) modifications to
BH entropy of a $5$D RNBH. We find that while the case is so for larger
(classical) geometries, the situation is radically altered for smaller
(quantum) geometries. Namely, we show surprising emergence of multiple phase
transitions that depend on the choice of extent of corrections to BH entropy
and charge. Our consideration involves differentiated extremal and non-extremal
geometric scales corresponding to the validity regime of corrections to
entropy. More emphasis is laid on the exponential case as the contributions
become highly non-trivial on small scales. An essential critical mass scale
arises in this case that marks the onset of these phase transitions while the
BH diminishes in size via Hawking evaporation. We contend that this critical
value of mass perhaps best translates as the epoch of a classical to quantum BH
phase transition.
