Thermodynamics of Four-Dimensional Einstein--Gauss--Bonnet Black Holes with a Modified Temperature

We compare two thermodynamic prescriptions associated with the same static, spherically symmetric four-dimensional Einstein--Gauss--Bonnet anti-de Sitter black hole. Model A uses the usual Hawking temperature obtained from the surface gravity and consequently yields an entropy with a logarithmic correction. Model B preserves the shift symmetry of the scalar field at the level of the thermodynamic charges, recovers the Bekenstein--Hawking area law, and requires a modified temperature for consistency with the first law. We derive the equations of state, Gibbs free energies, heat capacities, and critical quantities of both prescriptions in the extended phase space. The two models display the same qualitative Van der Waals phase structure, but their critical scales and local stability regions differ quantitatively. In the uncharged case, Model A has a larger critical horizon radius and lower critical temperature and pressure than Model B. A representative charged example further shows that the fixed ratios between the critical quantities of the two prescriptions in the neutral sector do not persist when $Q\\neq0$.