  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 $k=-1$.
