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

In this paper, we present two new families of spatially homogeneous black hole solution for $z=4$ Hov{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 are considered to have Bianchi types $II$ and $III$ symmetries, and hence the horizons are modeled on two types of Thurston $3$-geometries, namely the Nil geometry and $H^2times R$. Being foliated by compact 3-manifolds, the horizons are neither spherical, hyperbolic, nor toroidal, and therefore are not of the previously studied topological black hole solutions in Hov{r}ava-Lifshitz gravity. Using the Hamiltonian formalism, we establish the conventional thermodynamics of the solutions defining the mass and entropy of the black hole solutions for several classes of solutions. It turned out that for both horizon geometries the area term in the entropy receives two non-logarithmic negative corrections proportional to Hov{r}ava-Lifshitz parameters. Also, we show that choosing some proper set of parameters the solutions can exhibit locally stable or unstable behavior.