Restricted Phased Space Thermodynamics for Black Holes in Higher Dimensions and Higher Curvature Gravities

The recently proposed restricted phase space thermodynamics is shown to be applicable to a large class of higher dimensional higher curvature gravity models coupled to Maxwell field, which are known as black hole scan models and are labeled by the spacetime dimension <i>d</i> and the highest order <i>k</i> of the Lanczos-Lovelock densities appearing in the action. Three typical example cases with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>d</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> are chosen as example cases and studied in some detail. These cases are representatives of Einstein-Hilbert, Chern-Simons and Born-Infield like gravity models. Our study indicates that the Einstein-Hilbert and Born-Infield like gravity models have similar thermodynamic behaviors, e.g., the existence of isocharge <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> phase transitions with the same critical exponents, the existence of isovoltage <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> transitions and the Hawking-Page like transitions, and the similar high temperature asymptotic behaviors for the isocharge heat capacities, etc. However, the Chern-Simons like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>-model behaves quite differently. Neither isocharge nor isovoltage <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> transitions could occur and no Hawking-Page like transition is allowed. This seems to indicate that the Einstein-Hilbert and Born-Infield like models belong to the same universality class while the Chern-Simons like models do not.