Thermal superpotential and thermodynamics of neutral hairy black holes in extended SUGRA

A bstract We present a family of exact neutral hairy black-hole solutions with spherical horizon topology in extended supergravity with Fayet-Iliopoulos terms. We consider a consistent dilaton truncation and analyze in detail a sector where the magnetic part of the FI terms vanishes. Using appropriate dilaton counterterms, we compute the thermodynamic quantities and show the existence of Hawking-Page phase transitions. As a holographic application, we derive the thermal superpotential in closed form and use it as a counterterm, explicitly demonstrating that no additional finite counterterms are required to regularize the Euclidean action and the quasi-local stress tensor. The dual stress tensor matches that of a thermal gas of massless particles and is consistent with mixed dilaton boundary conditions that preserve conformal symmetry.