  We present for the first time a Friedmann-like construction in the framework
of an osculating Finsler-Randers-Sasaki geometry. In particular, we consider a
vector field in the metric on a Lorentz tangent bundle, and thus the curvatures
of horizontal and vertical spaces, as well as the extra contributions of
torsion and non-linear connection, provide an intrinsic richer geometrical
structure, with additional degrees of freedom, that lead to extra terms in the
field equations. Applying these modified field equations at a cosmological
setup we extract the generalized Friedmann equations for the horizontal and
vertical space, showing that we obtain an effective dark energy sector arising
from the richer underlying structure of the tangent bundle. Additionally, as it
is common in Finsler-like constructions, we obtain an effective interaction
between matter and geometry. Finally, we consider a specific model and we show
that it can describe the sequence of matter and dark-energy epochs, and that
the dark-energy equation of state can lie in the quintessence or phantom
regimes, or cross the phantom divide.
