  Horava gravity has been proposed as a renormalizable, higher-derivative,
Lorentz-violating quantum gravity model without ghost problems. A Horava
gravity based dark energy (HDE) model for dynamical dark energy has been also
proposed earlier by identifying all the extra (gravitational) contributions
from the Lorentz-violating terms as an effective energy-momentum tensor in
Einstein equation. We consider a complete CMB, BAO, and SNe Ia data test of the
HDE model by considering general perturbations over the background perfect HDE
fluid. Except from BAO, we obtain the preference of non-flat universes for all
other data-set combinations. We obtain a positive result on the cosmic tensions
between the Hubble constant H0 and the cosmic shear S8, because we have a shift
of H0 towards a higher value, though not enough for resolving the H0 tension,
but the value of S8 is unaltered. This is in contrast to a rather decreasing H0
but increasing S8 in a non-flat LCDM. For all other parameters, like Omega_m
and Omega_Lambda, we obtain quite comparable results with those of LCDM for all
data sets, especially with BAO, so that our results are close to a cosmic
concordance between the datasets, contrary to the standard non-flat LCDM. We
also obtain some undesirable features, like an almost null result on Omegak,
which gives back the flat LCDM, if we do not predetermine the sign of Omegak,
but we propose several promising ways for improvements by generalizing our
analysis.
