Abstract In this work, we present a new spherically symmetric black hole solution surrounded by a King-type dark matter halo. The construction begins by formulating the spacetime metric corresponding to a pure dark matter distribution, utilizing the established relationship between the tangential velocity in the equatorial plane and the metric coefficients of a spherically symmetric geometry. By incorporating the King dark matter profile into the energy-momentum tensor within Einstein’s field equations, we obtain an exact black hole solution describing the coupled system. To investigate the influence of the halo parameters-specifically, the core density and core radius-on the optical properties of the black hole, we analyze null geodesics within the Lagrangian framework. The resulting effective potential, photon sphere, and shadow structures are systematically examined. Notably, the shadow radius is found to exceed the corresponding Schwarzschild value, displaying a significant increase as the core radius of the King profile grows. Finally, we analyze the thermodynamic properties of the obtained black hole solution and found that the dark matter halo parameters strongly influence its thermodynamic behavior. These parameters significantly affect the Hawking temperature, Gibbs free energy, and specific heat capacity, thereby shaping the black hole’s stability and phase structure. Furthermore, within the framework of thermodynamic topology, our analysis of the generalized free energy shows a topological charge W = -1 W = - 1, indicating that the black hole shares the same topological class as the Schwarzschild case and possesses a single thermodynamically stable phase without multiple phase transitions in the explored regime.
Al-Badawi et al. (Wed,) studied this question.