In numerous applications, the reliability of a numerical approximation hinges not only on accuracy but also on the preservation of crucial structural properties inherent to the equation and its physical meaning. These properties may include positivity or long-time behaviour of the solutions, as well as the existence of specific steady states. Here, we focus on the approximation of advection-diffusion equations, which are a base block for many dissipative systems. We introduce an arbitrary-order (in space) nonlinear scheme based on the Hybrid High-Order (HHO) methodology, which can handle unconditionally anisotropic diffusion and general polyhedral meshes. The method is designed to provide positive solutions while ensuring correct discrete long-time behaviour. The key feature of the scheme lies in the discrete-level preservation of an entropy structure, mimicking its continuous counterpart. Numerical evidence is presented to point out the robustness of our method, as well as to investigate its usefulness with respect to low-order schemes.