This work develops the large deviations theory for the point process associated with the Euclidean and hyperbolic volume of k-nearest neighbor (k-NN) balls. In the Euclidean setting, the k-NN balls are centered around the points of a homogeneous Poisson process in the torus. In the hyperbolic setting, we consider a stationary Poisson point process of unit intensity in a growing sampling window in the hyperbolic space. Two different types of large deviation behaviors are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-NN balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes. Joint work with Christian Hirsch, Taegyu Kang, Moritz Otto, and Christoph Th\"ale.