We analyze the approximation of the acoustic wave equation in its first-order Friedrichs formulation by explicit Runge-Kutta (ERK) schemes in time combined with hybrid high-order (HHO) methods in space. We propose two general assumptions (I1)-(I2) for an interpolation operator to evaluate the consistency error, and establish energy-error estimates in the time-continuous setting. We give several examples of interpolation operators: the classical one in the HHO literature based on L2-orthogonal projections and others from, or inspired from, the hybridizable discontinuous Galerkin (HDG) literature giving improved convergence rates on simplices. In the fully discrete analysis, the key observation is that it becomes crucial to bound the consistency error in space by means of the stabilization seminorm only. We formulate three abstract properties (A1)-(A3) to lead the analysis and prove that, under suitable CFL conditions for second-and third-order ERK schemes, the energy error converges optimally in time and quasi-optimally in space, with optimal rates recovered on simplicial meshes. The abstract foundations of our analysis should facilitate its application to other nonconforming hybrid methods such as HDG and weak Galerkin (WG) methods.