In this work, we introduce and analyze a hp-Hybrid High-Order method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We formulate hp-convergence estimates for both the energy-and L2-norms of the error, which are the first results of this kind for Hybrid High-Order methods. The estimates are fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on its (local) anisotropy. The expected exponential convergence behaviour is numerically shown on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.