Senjaya D.Klaypong A.Mahidol University2026-04-162026-04-162026-03-01European Physical Journal C Vol.86 No.3 (2026)14346044https://repository.li.mahidol.ac.th/handle/123456789/116224We investigate the thermodynamics and phase structure of an extremely rare exact, static and spherically symmetric black hole solution in the Einstein–Skyrme theory. The solution is characterized by two matter-sector couplings: K, which controls the strength of the Skyrme field and induces a solid-angle-deficit-like factor 1-8πK in the metric and λ, the coupling of the quartic Skyrme term, which generates an r-2 contribution analogous to an effective charge. We formulate a novel first law of black hole thermodynamics by treating both K and λ as extensive thermodynamic variables. Within this framework, we derive the Hawking temperature, Gibbs free energy and heat capacity and analyze the resulting phase structure in the G-T and CH-T planes. Using an implicit method, we uncover Van der Waals-like critical behavior and associated phase transitions. We show that the Skyrme couplings substantially modify the thermodynamic behavior, play a crucial role in governing black hole stability and allow for the existence of phase transitions.Physics and AstronomyEngineeringArticleSCOPUS10.1140/epjc/s10052-026-15569-42-s2.0-10503496886314346052