Deformed Hořava–Lifshitz cosmology and stability of the Einstein static universe

We investigate the stability of the Einstein static universe under linear scalar, vector, and tensor perturbations in the context of a deformed Hořava-Lifshitz {(}HL{\rm)} cosmology related to entropic forces. We obtain a general stability condition under linear scalar perturbations. Using this general condition, we show that there is no stable Einstein static universe in the case of a flat universe $(k=0)$. In the special case of large values of the parameter $\omega$ of HL gravity in a positively curved universe $(k>0)$, the domination of the quintessence and phantom matter fields with a barotropic equation of state parameter $\beta<-1/3$ is necessary, while for a negatively curved universe $(k<0)$, matter fields with $\beta>-1/3$ must be the dominant fields of the universe. We also demonstrate a neutral stability under vector perturbations. We obtain an inequality including the cosmological parameters of the Einstein static universe for stability under tensor perturbations. It turns out that for large values of $\omega$, there is stability under tensor perturbations.