The standard Riemannian cosmological models studied by various authors, which assume isotropy, often lead to singularities and fail to account for directional dependencies in the early universe. In this paper we consider Finslerian approach is employed to introduce anisotropy into the spacetime structure, thereby enabling a richer and more realistic modeling of cosmic evolution. In this work, we consider an anisotropic Bianchi type-III model of the universe within the framework of generalized Finsler–Randers spacetime with \(f(\mathcal {R},\mathcal {T})\) gravity using the osculating Riemannian approach. This spacetime represents a specific class of Finsler geometry, which generalizes Riemannian geometry by allowing the metric to depend on both spacetime coordinates and a directional vector field. Next, we derive the field equations of \(f(\mathcal {R},\mathcal {T})\) gravity with Finsler–Randers Bianchi type-III model of universe including the cosmological constant \(\Lambda \) and obtain the corresponding modified Friedmann equations characterized by the presence of the Randers anisotropy function \(b_0(t)\) . Also, we derive an exact analytical solution of the modified Friedmann equations and evaluate various cosmological parameters. In addition, the Raychaudhuri equation is analyzed within the generalized Finsler–Randers framework. Finally, we discuss the physical implications of the model and provide a comparison with the \(\Lambda \) CDM cosmological model.
