In this paper, we explore static, spherically symmetric hairy black holes constructed through the extended gravitational decoupling method within a quantum spacetime endowed with a radially dependent noncommutative parameter. This setup naturally bridges the gap between a quantum-smeared interior and an asymptotic classical exterior. The hair is introduced via a decoupling parameter θ, leading to an exact metric determined by a specific integral-based mass profile. By solving the horizon equation A(r h ) = θ and deriving the Hawking temperature from the surface gravity, we uncover a rich thermal structure where the competition between geometric deformation and radially varying smearing generates multi-branch behavior. We further analyze the Bekenstein-Hawking entropy and assess stability through heat capacity and Helmholtz free energy. Complementing the thermodynamic analysis, we investigate the optical properties of the solution by deriving the effective potential for null geodesics and calculating the black hole shadow radius. Our results reveal that the dynamical quantum structure modifies the photon sphere and leads to a distinctive enlargement of the shadow silhouette in the celestial plane compared to standard noncommutative models. Overall, our findings indicate that radially dependent noncommutativity significantly enriches the phase diagram, offering regions of enhanced thermodynamic stability and pointing toward the possibility of stable remnants in specific parameter regimes.
Mansour et al. (Wed,) studied this question.
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