The implications of thermal geometries on Brane-World black hole (consisting of cosmological and dark matter effects) will be investigated by extracting the stability conditions (small and large roots) and divergences. To achieve this, we develop the temperature and heat capacity with respect to horizon radius, cosmological and dark matter parameters, respectively, which help in analyzing the small and large roots (stability conditions) and divergences. Furthermore, we study various methods of thermodynamical geometry such as Weinhold, Ruppeiner, Hendi–Panahiyah–Eslam–Momennia (HPEM), and Geometrothermodynamics (GTD) formulations and derive corresponding scalar curvatures for Brane-World black hole. It is found that Ruppeiner and HPEM formulations provide more information as compared to Weinhold and GTD methods for Brane-World black holes.

将通过提取稳定性条件（小根和大根）和散度来研究热几何学对Brane-World黑洞的影响（包括宇宙学和暗物质效应）。为此，我们分别针对视界半径，宇宙学和暗物质参数开发了温度和热容量，这有助于分析小根和大根（稳定性条件）和散度。此外，我们研究了各种热力学几何方法，例如Weinhold，Ruppeiner，Hendi-Panahiyah-Eslam-Momennia（HPEM）和地热力学（GTD）公式，并得出了Brane-World黑洞的相应标量曲率。发现与用于Brane-World黑洞的Weinhold和GTD方法相比，Ruppeiner和HPEM配方提供了更多信息。