We prove the optimal convergence in space and time for the linear acoustic wave equation in its second-order formulation in time, using the hybrid high-order method for space discretization and the leapfrog (central finite difference) scheme for time discretization. The proof hinges on energy arguments similar to those classically deployed in the context of continuous finite elements or discontinuous Galerkin methods, but some novel ideas need to be introduced to handle the static coupling between cell and face unknowns. Because of the close ties between the methods, the present proof can be readily extended to cover space semi-disretization using the hybridizable discontinuous Galerkin method and the weak Galerkin method.