Thermodynamic analysis of black holes with cloud of strings and quintessence via Barrow entropy

We explore a Reissner-Nordstr\"{o}m Anti-de Sitter (RN-AdS) black hole with a cloud of string and quintessence to study thermodynamics and thermodynamic topology in the presence of Barrow entropy, which is currently being used widely as the horizon of a black hole may not be a smooth surface as described in classical general relativity but instead could have a more intricate fractal-like structure. Here, we study the impact of the fractal correction parameter of Barrow entropy on the thermodynamics of such BHs. We compute the first law of black hole thermodynamics and the Smarr relation for RN-AdS black hole with a cloud of string and quintessence in terms of Barrow entropy by employing the generalized formula for spherically symmetric spacetime which is directly derived from the Einstein field equation. The significance of Barrow entropy has been verified from thermodynamic topology as well. We also found that the non-zero topological charge indicates the presence of the critical point.