Thermodynamic and Phase Structure of a Bardeen Black Hole Surrounded by a Cloud-of-Strings and Quintessence

This paper examines the thermal properties of a Bardeen-type black hole (BH) embedded in a cloud-of-strings and a quintessence background within the Tsallis non-extensive entropy framework. Within the extended thermodynamic phase space, generalized expressions are derived for the Tsallis thermodynamic temperature, heat capacity, equation of state, Gibbs free energy, and the associated critical parameters. Particular attention is paid to how the cloud-of-strings parameter, the magnetic charge and the quintessence field influence the resulting thermodynamic behaviour. A divergence in the heat capacity marking the boundary of local thermodynamic stability is identified from the heat-capacity analysis. The equation of state is further examined using the standard inflection-point conditions. For the representative parameter set considered in this work, the resulting mathematical critical solution has negative critical temperature and pressure and therefore lies outside the physical thermodynamic domain. Hence, no physically admissible Van der Waals-type critical point is obtained for the chosen parameters. Furthermore, the thermodynamic geometry is investigated using the HPEM metric, whose Ricci scalar provides an independent geometric characterization of the phase-transition structure. The results demonstrate that the cloud-of-strings parameter, magnetic charge, and quintessence field significantly influence the thermodynamic stability and phase structure of the BH, whereas the Tsallis entropy modifies the quantitative thermodynamic behaviour while preserving the overall qualitative phase characteristics.