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In this paper, we present two new families of spatially homogeneous blackhole solution for $z=4$ Ho\v{r}ava-Lifshitz Gravity equations in $(4+1)$dimensions with general coupling constant $\lambda$ and the especial case$\lambda=1$, considering $\beta=-1/3$. The three-dimensional horizons areconsidered to have Bianchi types $II$ and $III$ symmetries, and hence thehorizons are modeled on two types of Thurston $3$-geometries, namely the Nilgeometry and $H^2\times R$. Being foliated by compact 3-manifolds, the horizonsare neither spherical, hyperbolic, nor toroidal, and therefore are not of thepreviously studied topological black hole solutions in Ho\v{r}ava-Lifshitzgravity. Using the Hamiltonian formalism, we establish the conventionalthermodynamics of the solutions defining the mass and entropy of the black holesolutions for several classes of solutions. It turned out that for both horizongeometries the area term in the entropy receives two non-logarithmic negativecorrections proportional to Ho\v{r}ava-Lifshitz parameters. Also, we show thatchoosing some proper set of parameters the solutions can exhibit locally stableor unstable behavior.

Spatially homogeneous black hole solutions in $z=4$ Ho\v{r}ava-Lifshitz gravity in $(4+1)$ dimensions with Nil geometry and $H^2\times R$ horizons