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Thermodynamic Riemannian geometry provides great insights into themicroscopic structure of black holes (BHs). One such example is the Ruppeinergeometry which is the metric space comprising the second derivatives of entropywith respect to other extensive variables of the system. Reissner-Nordstr\"omblack holes (RNBHs) are known to be endowed with a flat Ruppeiner geometry forall higher spacetime dimensions. However this holds true if one invokesclassical gravity where the semi-classical Bekenstein-Hawking entropy bestdescribes the thermodynamics of the system. If the much deeper quantum gravityand string theories entail modifications to BH entropy, this prompts thequestion whether the Ruppeiner flatness associated with higher dimensionalRNBHs still persists. We investigate this problem by consideringnon-perturbative (exponential) and perturbative (logarithmic) modifications toBH 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 phasetransitions that depend on the choice of extent of corrections to BH entropyand charge. Our consideration involves differentiated extremal and non-extremalgeometric scales corresponding to the validity regime of corrections toentropy. More emphasis is laid on the exponential case as the contributionsbecome highly non-trivial on small scales. An essential critical mass scalearises in this case that marks the onset of these phase transitions while theBH diminishes in size via Hawking evaporation. We contend that this criticalvalue of mass perhaps best translates as the epoch of a classical to quantum BHphase transition.

On the microstructure of higher-dimensional Reissner-Nordstr\"om black holes in quantum regime