Thermodynamic stability and geometric thermodynamics of charged black holes in Rastall-massive gravity under quintessence
In the present paper, we derive a new charged black hole solution embedded in a quintessence field within a framework combining the Rastall and massive gravity in an asymptotically flat spacetime. We interpret the integration constant associated with the structural properties of the quintessence field as a thermodynamical variable. In this context, several limiting cases have been explored with the aim to shed light on the role of the model parameters. Then, we analyse the thermodynamic properties and stability of the charged black holes in flat spacetime, where the quintessence field is characterized by the state parameter \( \omega_q = -2/3 \). Detailed studies are perfomed to see how the various parameters affect phase transitions between small and large black holes. In particular, we establish a connection between the impact parameter and unstable circular photon orbits with the thermodynamic phase transitions. Our analysis suggest that the photon sphere observables could remarkably provide an indirect observational signature of black hole thermodynamic critical transitions. Generally, thanks to the effective contribution of the quintessence parameter, we found that the overall thermodynamic behavior basically resembles that of the charged RN--AdS black hole with thermal features analogous to of a Van der Waals fluid ones. Finally, we have checked these findings using thermodynamic geometry with the Quevedo metric. Our results showed that the critical behaviour derived from the standard thermodynamics is fully consistent with geothermodynamics description, where the phase structure and divergence points exactly coincide with those identified from heat capacity and Gibbs free energy analyses.