Hawking-Page phase transition for pure Lovelock black holes

We investigate the thermodynamic properties of static, spherically symmetric Anti-de Sitter (AdS) black holes, focusing on the interplay between characteristic temperatures, as well as on the universality of Ruppeiner scalar curvature at the Hawking-Page (HP) phase transition. In particular, we study the relation between the minimum temperature and the HP phase transition temperature for static, spherically symmetric AdS black holes in pure Lovelock gravity. For the electromagnetically neutral case in Einstein gravity, the minimum temperature in (d+1) dimensions coincides with the HP transition temperature in d dimensions, while in higher pure Lovelock theories this relation is modified by a dimension- and order-dependent factor, reducing to the Einstein result in appropriate limits. For charged AdS black holes, in the grand canonical ensemble, in general relativity, the two temperatures differ by a simple dimension-dependent factor, whereas no universal relation persists in higher curvature pure Lovelock theories. We further analyze the normalized Ruppeiner scalar curvature at the HP transition and show that it is a universal constant depending only on the spacetime dimension for electromagnetically neutral black holes in pure Lovelock theories. The normalized scalar curvature remains a constant, under appropriate conditions, even for the charged static spherically symmetric black holes in the grand canonical ensemble for the Einstein theory case, whereas in general pure Lovelock theories it depends on thermodynamic parameters such as pressure and electrostatic potential, asymptotically approaching a constant in the large-pressure or simultaneous large-potential and large-pressure limits.