Using the dynamical system approach, properties of cosmological models based on the Hořava-Lifshitz gravity are systematically studied. A result of this investigation is that in the detailed balance case one of the attractors in the theory corresponds to the oscillatory behavior described by Brandenberger. Instead the cosmological models generated by Hořava-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.
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