Thermodynamics phase transition of Anti de Sitter Schwarzschild scalar-tensor-vector-Black Holes

Instead of scalar tensor gravity models which is applicable for description of cosmic inflation with unknown dark sector of matter/energy, at presentense there are presented different alternative scalar tensor vector gravities where meaningful dynamical vector fields can support cosmic inflation well without to use dark matter/energy concept. One of these gravity models was presented by Moffat which its modified Schwarzschild black hole solution is used to study thermodynamic phase transition in presence of the AdS space pressure in this article. To do so we obtained an equation of state which asymptotically reaches to equation of state of ideal gas for large black holes but for small scale black holes we obtained a critical point at phase space where the black hole can be exhibit with a phase transition at processes of isotherm and isobaric. By looking at diagrams of the Gibbs free energy and the heat capacity at constant pressure which are plotted versus the temperature and the specific volume one can see an inflection point which means that the phase transition is type of second order. In fact there is small to large phase transition for the black hole which is equivalent to the Van der Waals liquid-gas phase transition in ordinary thermodynamic systems. The phase transition is happened below the critical point in phase space when the gravitational charge of the black hole is equal to its mass.Comment: 21 pages, 4 figure