Maxwell construction and multi-criticality in uncharged generalized quasi-topological black Holes

We demonstrate the existence of $N$-tuple critical points of uncharged AdS black holes in generalized quasi-topological theories. The criticality is shown to have a geometrical interpretation described by the Maxwell's equal area rule. We present a compact reformulation of the area rule and identify a criterion for the emergence such points. Using this criterion, we construct several multi-critical points with genuine generalized quasi-topological densities, including a quadruple and a quintuple points.