  We further investigate the dark energy model based on the Finsler geometry
inspired osculating Barthel-Kropina cosmology. The Barthel-Kropina cosmological
approach is based on the introduction of a Barthel connection in an osculating
Finsler geometry, with the connection having the property that it is the
Levi-Civita connection of a Riemannian metric. From the generalized Friedmann
equations of the Barthel-Kropina model, obtained by assuming that the
background Riemannian metric is of the Friedmann-Lemaitre-Robertson-Walker
type, an effective geometric dark energy component can be generated, with the
effective, geometric type pressure, satisfying a linear barotropic type
equation of state. The cosmological tests, and comparisons with observational
data of this dark energy model are considered in detail. To constrain the
Barthel-Kropina model parameters, and the parameter of the equation of state,
we use 57 Hubble data points, and the Pantheon Supernovae Type Ia data sample.
The st statistical analysis is performed by using Markov Chain Monte Carlo
(MCMC) simulations. A detailed comparison with the standard $\Lambda$CDM model
is also performed, with the Akaike information criterion (AIC), and the
Bayesian information criterion (BIC) used as the two model selection tools. The
statefinder diagnostics consisting of jerk and snap parameters, and the $Om(z)$
diagnostics are also considered for the comparative study of the
Barthel-Kropina and $\Lambda$CDM cosmologies. Our results indicate that the
Barthel-Kropina dark energy model gives a good description of the observational
data, and thus it can be considered a viable alternative of the $\Lambda$CDM
model.
