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In this work, we investigate thermodynamic properties and Hawking-Page phasetransition of a black hole in the Schwarzschil-Anti-de Sitter-Beltrami (SAdSB)spacetime. We discuss the Beltrami or inertial coordinates of the Anti-deSitter(AdS) spacetime. Transformation between the non-inertial and inertialcoordinates of the AdS spacetime is formulated to construct the solution ofspherical gravitating mass and other physical quantities. The Killing vector isdetermined to calculate the event horizon radius of this black hole. Theentropy and the temperature of SAdSB black hole are determined by the Noethercharge method and it is shown that the temperature is bounded by the Anti-deSitter radius. Similarly, the Smarr relation and first law of black holethermodynamics for the SAdSB spacetime have been formulated. The Gibbs freeenergy and heat capacity of this black hole are calculated and we consider thephase transition between small and large black holes. The first-order phasetransition between the thermal AdS spacetime and large black hole phase is alsoinvestigated and the Hawking-Page temperature is computed and compared with thecase of the Schwarzschild-Anti-de Sitter(SAdS) black hole.

Noether Charge, Thermodynamics and Phase Transition of a Black Hole in the Schwarzschild-Anti-de Sitter-Beltrami spacetime