This work investigates the thermodynamic behavior and Joule–Thomson (JT) expansion of Hayward-Letelier Anti-de Sitter (HL-AdS) black holes (BHs). Our analysis is framed within the recently established universal topological classification of BH thermodynamics, where all BHs can be categorized into four distinct topological classes: \( W^{1-} \) , \( W^{0+} \) , \( W^{0-} \) , and \( W^{1+} \) . For small charge parameters, the system exhibits van der Waals-like criticality; however, we find that increasing the charge parameter leads to a regime of global stability regardless of BH size. By employing the Duan mapping topological current theory, we determine that the system is characterized by a total topological charge of \(Q = -1\) , categorizing it within the \(W^{1+}\) universal class. This suggests that the ’regularity’ of the Hayward core is topologically robust and remains fundamentally unchanged by the presence of a background string density. The existing BH models that do not have singularity problems show effective solutions for central curvature singularities; how Letelier cloud string topological defects affect their thermodynamic stability remains largely unexplored. We also adopt the geometrothermodynamics method in order to explore the thermodynamic properties and the phase structure of the HL-AdS BHs while simultaneously considering nonlinear magnetic charge and the presence of a string cloud. This research examines how external string structures interact with the internal topological landscape of the system to assess whether the regular core can maintain stability against different types of gravitational fields.
