Dual relations from Hawking-Page transition in Gauss-Bonnet gravity and Gauss-Bonnet-Maxwell gravity

A universal dual relation $T_0(n+1)=T_{HP}\left(n\right)$ between the minimum temperature ($T_0(n+1)$) black hole phase and Hawking-Page (HP) transition ($T_{HP}\left(n\right)$) black hole phase in two successive dimensions was introduced in [1], which was reminiscent of the AdS/CFT correspondence, as the HP transition temperature could be treated as the temperature of the dual physical quantity on the boundary and the latter corresponds to the one in the bulk. In this paper, we derive analytically the dual relations in Gauss-Bonnet (GB) gravity and GB-Maxwell gravity. For the GB (charged/uncharged) spherical AdS black holes, the dual relations are exactly the same with the ones in Einstein gravity. Especially for the GB hyperbolic AdS black holes, since there exist the HP transition with reentrance and triple points, the dual relations only hold while they characterize actually the duality between the (large/small) HP transition temperature and the extremum (minimum/maximum) temperature in two successive dimensions. In the grand canonical ensemble of GB gravity, the Gibbs energy has the similar qualitative behavior with the cases in the canonical ensemble of GB gravity, while there is an additional effect of electric potential on the HP transition. These dual relations are interesting in understanding the HP transition, and may bring some clue on the applications in the holographic principle and the black hole thermodynamics.