| Summary: | This paper presents an exact black hole solution within a cloud of strings (CoS) and a perfect fluid dark matter (PFDM) field. The derived black hole solution interpolates between the AdS-charged Letelier black hole in the absence of the PFDM field and the AdS Reissner-Nordström black hole in the limiting case of the PFDM and CoS parameters. The thermodynamic properties of the black hole, including temperature, entropy, heat, and free energy, are modified in the presence of the PFDM field and CoS parameter. Furthermore, we investigate the black hole's critical points and phase structure in an extended phase space. We find that the critical temperature and pressure decrease with the CoS parameter but increase with the PFDM field. The Gibbs free energy versus temperature plot reveals a swallow-tail behavior, indicating a first-order phase transition, which terminates at a second-order phase transition.
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