On the role of boundary conditions within Ho˘ rava-Lifshitz gravity Lissa de Souza Campos 1,2 , ∗ Claudio Dappiaggi 1,2 , † and Denis Sina 1‡ 1 Dipartimento di Fisica, Università degli Studi di Pavia, Via Bassi, 6, 27100 Pavia, Italy and 2 Istituto Nazionale di Fisica Nucleare – Sezione di Pavia, Via Bassi, 6, 27100 Pavia, Italy (Dated: March 30, 2021) Abstract On a class of four-dimensional Lifshitz spacetimes with critical exponent z =2, including a hyperbolic and a spherical Lifshitz topological black hole, we consider a real Klein-Gordon field. Using a mode-decomposition, we split the equation of motion into a radial and into an angular component. As first step, we discuss under which conditions on the underlying parameters, we can impose to the radial equation boundary conditions of Robin type and whether bound state solutions do occur. Subsequently, we show that, whenever bound states are absent, one can associate to each admissible boundary condition a ground and a KMS state whose associated two-point correlation function is of local Hadamard form. Keywords: Klein-Gordon equation, Robin boundary conditions, Lifshitz black holes. * lissa.desouzacampos01@universitadipavia.it † claudio.dappiaggi@unipv.it ‡ denis.sina01@universitadipavia.it 1 arXiv:2103.15391v1 [hep-th] 29 Mar 2021