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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 $Λ$CDM model is also performed, and it turns out that the Barthel-Kropina type models give a satisfactory description of the observations.