Using the dynamical system approach, the properties of cosmological models based on the Hoava-Lifshitz gravity are systematically studied. In particular, the cosmological phase space of the Hoava-Lifshitz model is characterized. The analysis allows us to compare some key physical consequences of the imposition (or not) of detailed balance. The result of the investigation is that in the detailed balance case one of the attractors in the theory corresponds to an oscillatory behavior. Such oscillations can be associated with a bouncing universe, as previously described by Brandenberger, and will prevent a possible evolution toward a de Sitter universe. Other results obtained show that the cosmological models generated by Hoava-Lifshitz gravity without the detailed balance assumption have indeed the potential to describe the transition between the Friedmann and the dark-energy eras. The whole analysis leads to the plausible conclusion that a cosmology compatible with the present observations of the universe can be achieved only if the detailed balance condition is broken. © 2010 IOP Publishing Ltd.