  We consider a very general scenario of our universe where its geometry is
characterized by the Finslerian structure on the underlying spacetime manifold,
a generalization of the Riemannian geometry. Now considering a general
energy-momentum tensor for matter sector, we derive the gravitational field
equations in such spacetime. Further, to depict the cosmological dynamics in
such spacetime proposing an interesting equation of state identified by a sole
parameter $\gamma$ which for isotropic limit is simply the barotropic equation
of state $p= (\gamma- 1) \rho$ ($\gamma \in \mathbb{R}$ being the barotropic
index), we solve the background dynamics. The dynamics offers several
possibilities depending on this sole parameter as follows $-$ (i) only an
exponential expansion, or (ii) a finite time past singualrity (big bang) with
late accelerating phase, or (iii) a nonsingular universe exhibiting an
accelerating scenario at late time which finally predicts a big rip type
singularity. We also discuss several energy conditions and the possibility of
cosmic bounce. Finally, we establish the first law of thermodynamics in such
spacetime.
