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