Thermodynamic Topology of $D=4,5$ Horava Lifshitz Black Hole in Two Ensembles

We study the thermodynamic topology of four and five dimensional Horava Lifshitz (HL) black holes in Horava gravity. These exotic black hole solutions belong to a special class of of black holes whose thermodynamics exhibit a line of (continuous) second order phase transitions known as $\lambda$ phase transitions akin to those observed in the superfluidity of liquid $^{4}He$. To analyze their thermodynamic topology, we treat the Horava Lifshitz (HL) black holes as topological defects in their thermodynamic spaces and compute the winding numbers at those defects. We work in two different ensembles: fixed $\epsilon$ ensemble and fixed $\zeta$ ensemble, where $\epsilon$ is a parameter of the HL black holes and $\zeta$ is its conjugate parameter. In the fixed $\epsilon$ ensemble, three different horizon types are considered : the spherical horizon for $k=+1$, the flat horizon for $k=0$, and the hyperbolic horizon for $k=-1$. In the fixed $\zeta$ ensemble, two different horizon types are considered : the spherical horizon for $k=+1$, and the hyperbolic horizon for $k=-1$.