  We use observational data from Type Ia Supernovae (SNIa), Baryon Acoustic
Oscillations (BAO), and Cosmic Microwave Background (CMB), along with
requirements of Big Bang Nucleosynthesis (BBN), to constrain the cosmological
scenarios governed by Horava-Lifshitz gravity. We consider both the detailed
and non-detailed balance versions of the gravitational sector, and we include
the matter and radiation sectors. We conclude that the detailed-balance
scenario cannot be ruled out from the observational point of view, however the
corresponding likelihood contours impose tight constraints on the involved
parameters. The scenario beyond detailed balance is compatible with
observational data, and we present the corresponding stringent constraints and
contour-plots of the parameters. Although this analysis indicates that
Horava-Lifshitz cosmology can be compatible with observations, it does not
enlighten the discussion about its possible conceptual and theoretical
problems.
