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At the present paper, it is studied cosmological solutions and its stabilityin the frame of F(R) Horava-Lifshitz gravity. The perturbations around generalspatially flat FRW solutions are analyzed and it is showed that the stabilityof those solutions will depend on the kind of theory, i.e. on the form of theaction F(R), as well as on the parameters contained in any Horava-Lifshitztheory due to the breaking of Lorentz invariance. The (in)stability of a givencosmic solution can restrict the models and gives new observationalpredictions, and can give a natural explanation on the end of inflation andradiation/matter phases. An explicit example of F(R) is studied, and it isshowed that the instability can produce the transition between the differentepochs of the Universe history.

Stability of cosmological solutions in F(R) Horava-Lifshitz gravity