Probing quantum gravity effects: Geodesic structure and thermodynamics of deformed Schwarzschild AdS black holes surrounded by cosmic strings

We investigate a deformed Schwarzschild anti-de Sitter black hole with a cloud of strings, performing a comprehensive analysis across three interconnected domains: geodesic structure, scalar perturbations, and thermodynamic properties. Our methodology incorporates three key parameters: the deformation parameter $\alpha$, the control parameter $\beta$, and the cosmic strings (CS) parameter $\zeta$. Through geodesic analysis, we determine the photon sphere radius and shadow cast, finding that $\zeta$ systematically increases these characteristic lengths, while $\alpha$ produces the opposite effect. For null geodesics, we derive the Lyapunov exponent and establish that the deformation reduces the orbital instability compared to the Letelier solution. Our investigation of scalar perturbations reveals how the combined effect of deformation and CS modifies the effective potential, with implications for black hole stability under external disturbances. The thermodynamic analysis demonstrates that deformation enables stable smaller black holes at lower temperatures, counteracting the destabilizing influence of CS. We observe a rich phase structure with three distinct regions: small black holes with negative heat capacity, intermediate black holes that are locally stable but globally unfavorable, and large black holes that achieve both local and global stability at sufficiently high temperatures. The Hawking–Page transition temperature increases with both $\alpha$ and $\zeta$, though through different underlying mechanisms. These results shed light on the collective effects of quantum gravity-motivated deformations and extended structures on black hole physics.