On thermodynamics of charged AdS black holes in extended phases space via M2-branes background

Motivated by a recent work on asymptotically Ad$$S_4$$ black holes in M-theory, we investigate both thermodynamics and the thermodynamical geometry of Reissner–Nordstrom-AdS black holes from M2-branes. More precisely, we study AdS black holes in $$AdS_{4}\times S^{7}$$, with the number of M2-branes interpreted as a thermodynamical variable. In this context, we calculate various thermodynamical quantities including the chemical potential, and examine their phase transitions along with the corresponding stability behaviors. In addition, we also evaluate the thermodynamical curvatures of the Weinhold, Ruppeiner, and Quevedo metrics for M2-branes geometry to study the stability of such a black object. We show that the singularities of these scalar curvature’s metrics reproduce similar stability results to those obtained by the phase transition diagram via the heat capacities in different ensembles either when the number of the M2 branes or the charge is held fixed. Also, we note that all results derived in Belhaj et al. (Eur Phys J C 76(2):73, 2016) are recovered in the limit of the vanishing charge