Thermodynamic topology of D = 4,5 Horava Lifshitz black hole in two ensembles

In this paper, we investigate the thermodynamic topology of four and five-dimensional Hořava-Lifshitz (HL) black holes within Hořava gravity. To analyze their thermodynamic topology, we treat these HL black holes as topological defects in their thermodynamic spaces and compute the winding numbers at these defects. Our study is conducted in two different ensembles: the fixed ϵ ensemble and the fixed ζ ensemble, where ϵ is a parameter of the HL black holes, and ζ is its conjugate parameter. In the fixed ϵ ensemble, we consider three different horizon types: spherical ($k=+1$), flat ($k=0$), and hyperbolic ($k=-1$), while in the fixed ζ ensemble, we examine two horizon types: spherical ($k=+1$) and hyperbolic ($k=-1$). For all the aforementioned cases, we study the local and global topology of these black holes and classify them into topological classes based on their topological charges. We calculate the topological number for each case. This process uncovers some novel phase diagrams that reveal a number of multiple defect curve phenomenon. Additionally, we explore how the topological charge of these black holes depends on the values of the thermodynamic parameters. Our findings indicate that the thermodynamic topology of Hořava-Lifshitz black holes in $D=4,5$ is significantly influenced by the type of horizon, the choice of ensemble, the values of pressure, and the parameters ϵ or ζ.