  We investigate the validity of a \emph{minimal} cosmological model derived
from the renormalizable Ho$\check{\textrm{r}}$ava action at low redshift scales
by using different cosmological and statistical tests. Assuming pure attractive
gravity, i.e., $\lambda>1/3$ in the Ho$\check{\textrm{r}}$ava action, we
compare the Union 2.1 supernova type Ia data with the kinematics following from
a model-independent approach. The two approaches, although compatible, lead to
explicit cosmographic constraints on the free parameters of the
Ho$\check{\textrm{r}}$ava action, which turn out to be in strong disagreement
with the $\Lambda$CDM, $w$CDM and Chevallier-Polarski-Linder scenarios. To show
this, we use standard diagnostic tools of regression models, namely the Akaike
and the Bayesian Information Criteria. Using such model-independent statistical
methods, we show that Ho$\check{\textrm{r}}$ava-Lifshitz cosmology differs from
the standard dark energy scenarios, \emph{independently} of the number of free
parameters involved in the model. Since this result is valid at small redshift
domains, it indicates the presence of inconsistencies in the minimal version of
Ho$\check{\textrm{r}}$ava-Lifshitz cosmology even at the level of background
cosmology.
