||It has been shown that electron-electron interactions (e-e) in Bernal-stacked multilayer graphene play a prominent role, leading the system into a gapped state at B = 0 for even-layered devices up to 8 layers. However, the origin of this state remains to be understood. Based on electrical transport measurements of high-quality suspended graphene devices, we use a novel approach to probe the low-energy density of states of graphene multilayer systems by studying the temperature dependence of the width of their resistance peak. This technique allowed us to directly observe a second-order electronic phase transition, which transition temperature (TC) increases linearly with increasing the thickness of the system, starting from 12K in bilayer up to 100K in heptalayer devices. We explain the origin of this transition with the incursion of a self-consistent valley- and spin-dependent staggered potential Δ, changing sign from one layer to the next. Our results accounts for the experimental observation of a finite-temperature phase transition driven by e-e interactions at B = 0 and provides additional constraints to the current microscopic theories which attempt to describe the interaction effects in multilayer graphene systems.