In this work, we investigate black hole solutions within the framework of Finsler–Randers geometry by employing the osculating Riemannian approach together with the Barthel connection. The anisotropic structure of spacetime is encoded through a linear Finslerian function \(\eta (r)=ar+b\) , representing weak geometric deviations from the Riemannian limit. Using this formulation, we derive exact static and spherically symmetric Finslerian black hole solutions, including the Schwarzschild Finsler black hole (SFBH) and its generalization in the presence of surrounding matter, the Kiselev Finsler black hole (KFBH). The Finslerian anisotropy modifies the metric functions and leads to shifted horizon structures compared to their general relativistic counterparts. We further analyze the thermodynamic properties of these black holes on the effective osculating Riemannian background, examining quantities such as temperature, entropy, effective mass, and heat capacity. The results demonstrate that Finslerian corrections introduce nontrivial modifications to black hole thermodynamics while remaining consistent with the standard laws in the appropriate limit. Our study provides a controlled and physically motivated framework for exploring anisotropic effects in black hole spacetimes and highlights the potential relevance of Finsler geometry in gravitational and cosmological settings.
