Abstract We explore the thermodynamics and geothermodynamics of black holes with the Barrow entropy in a brane-world scenario, where the horizon geometry of the black hole is regarded as a fractal structure. Our analysis reveals the behavior of heat capacity, identifying both bound and divergence points. For the Bekenstein-Hawking entropy, the divergence point exhibits smooth behavior, indicating no phase transition. In contrast, we observe divergence with Barrow entropy as the deformation parameter increases, confirming the presence of a zero point in heat capacity through various thermodynamic geometry formalisms. Additionally, we delve into thermodynamic topology, detailing the classification of black holes in the brane-world context and comparing their characteristics determined from the Bekenstein-Hawking and the Barrow entropy. Notably, fixing the deformation and cosmological parameters results in a topological charge -1 - 1 predominately by the dark matter parameter, which remains unaffected despite variations in other parameters. In the dS model, the cosmological horizon prevents stable photon spheres, making topological charges of 0 and +1 + 1 unattainable. Incremental increases in the cosmological parameter reduce the dark matter parameter-dominated region.
Zafar et al. (2026) studied this question.