Geodesics and thermodynamics of a $κ$-deformed anti-de Sitter black hole surrounded by a quintessence field

Abstract

We investigate the geodesic structure and thermodynamics of a $κ$-deformed Schwarzschild-anti-de Sitter black hole in a Kiselev quintessence field, where dynamics are governed by a mass-dependent lapse deformation. This term produces an inner Cauchy horizon without requiring charge or spin, yet it fails to resolve the central singularity; instead, the Kretschmann scalar diverges more strongly as $r^{-10}$. Because the deformation amplitude couples to the mass, the standard Bekenstein-Hawking area law violates the first law of thermodynamics. We resolve this by either deriving a modified entropy or rescaling the mass, both of which restore the first law and yield a consistent Smarr relation. Optically, an increasing deformation parameter shrinks both the photon sphere and the critical impact parameter. We also derive a closed-form Joule-Thomson coefficient, revealing that the $κ$-deformation triggers an inversion curve a feature absent in standard neutral SAdS. Notably, this occurs without generating a van der Waals critical point, demonstrating that Joule-Thomson inversion and van der Waals criticality can be cleanly decoupled. Finally, the deformation suppresses peak Hawking emission by a factor of roughly four and increases flux sparsity, an effect driven almost entirely by a drop in temperature rather than changes to the black hole shadow.