  In this article we present the cosmological equivalence between the
relativistic Finsler-Randers cosmology, with dark energy and modified gravity
constructions, at the background level. Starting from a small deviation from
the quadraticity of the Riemannian geometry, through which the local structure
of General Relativity is modified and the curvature theory is extended, we
extract the modified Friedmann equation. The corresponding extended
Finsler-Randers cosmology is very interesting, and it can mimic dark-energy and
modified gravity, describing a large class of scale-factor evolutions, from
inflation to late-time acceleration, including the phantom regime. In this
respect, the non-trivial universe evolution is not attributed to a new scalar
field, or to gravitational modification, but it arises from the modification of
the geometry itself.
