  Finsler geometry is an important extension of Riemann geometry, in which to
each point of the spacetime manifold an arbitrary internal variable is
associated. Interesting Finsler geometries, with many physical applications,
are the Randers and Kropina type geometries, respectively. A subclass of
Finsler geometries is represented by the osculating Finsler spaces, in which
the internal variable is a function of the base manifold coordinates only. In
an osculating Finsler geometry one introduces the Barthel connection, which has
the remarkable property that it is the Levi-Civita connection of a Riemannian
metric. In the present work we consider the gravitational and cosmological
implications of a Barthel-Kropina type geometry. We assume that in this
geometry the Ricci type curvatures are related to the matter energy-momentum
tensor by the standard Einstein equations. The generalized Friedmann equations
in the Barthel-Kropina geometry are obtained by considering that the background
Riemannian metric is of Friedmann-Lemaitre-Robertson-Walker type. The matter
energy balance equation is also derived. The cosmological properties of the
model are investigated in detail, and it is shown that the model admits a de
Sitter type solution, and that an effective dark energy component can also be
generated. Several cosmological solutions are also obtained by numerically
integrating the generalized Friedmann equations. A comparison of two specific
classes of models with the observational data and with the standard
$\Lambda$CDM model is also performed, and it turns out that the Barthel-Kropina
type models give a satisfactory description of the observations.
