A new pressure-based Lattice-Boltzmann method (HRR-p) is proposed for the simulation of flows for Mach numbers ranging from 0 to 1.5. Compatible with nearest neighbor lattices (e.g. D3Q19), the model consists of a predictor step comparable to classical athermal Lattice-Boltzmann methods, appended with a fully local and explicit correction step for the pressure. Energy conservation-for which the Hermi-tian quadrature is not accurate enough on such lattice-is solved via a classical finite volume MUSCL-Hancock scheme based on the entropy equation. The Euler part of the model is then validated for the transport of three canonical modes (vortex, en-tropy, and acoustic propagation), while its diffusive/viscous properties are assessed via thermal Couette flow simulations. All results match the analytical solutions, with very limited dissipation. Lastly, the robustness of the method is tested in a one dimensional shock tube and a two-dimensional shock-vortex interaction.