Hawking–Page transition and the dual relations of anti-de Sitter black holes surrounded by dark energy in general dimensions

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Recently, a dual relation T 0(n + 1) = T HP (n) between the minimum temperature (T 0(n + 1)) black hole phase and the Hawking–Page transition (T HP (n)) black hole phase in two successive dimensions was introduced by Wei et al (2020 Phys. Rev. D 102 10411); this was reminiscent of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, as the Hawking–Page transition temperature could be treated as the temperature of the dual physical quantity on the boundary and the latter corresponds to that in the bulk. In this paper, we discuss the Hawking–Page transition and the dual relations in AdS black holes surrounded by dark energy in general dimensions. Our findings reveal the occurrence of the Hawking–Page transition between the thermal AdS radiation and thermodynamically stable large AdS black holes, in both the spacetime surrounded by phantom dark energy and the spacetime surrounded by quintessence dark energy. We discuss the effects of the phantom dark energy and quintessence dark energy on the Hawking–Page transition temperature. For the dual relation in particular, it works well for the case of an AdS black holes surrounded by phantom dark energy. For the case of an AdS black hole surrounded by quintessence dark energy, the dual relation should be modified under an open assumption that the state parameter and the density parameter of the quintessence dark energy depend on the dimensions of the spacetime.