  We discuss the most general Finsler spacetime geometry obeying the
cosmological symmetry group $\mathrm{SO}(4)$. On this background geometry we
derive the equations of motion for the most general kinetic fluid obeying the
same cosmological symmetry. For this purpose we propose a set of coordinates on
the tangent bundle of the spacetime manifold which greatly simplifies the
cosmological symmetry generators.
