With growing interest in the simulation of compressible flows using the lattice Boltzmann (LB) method, a number of different approaches have been developed. These methods can be classified as pertaining to one of two major categories: (i) solvers relying on high-order stencils recovering the Navier–Stokes–Fourier equations, and (ii) approaches relying on classical first-neighbour stencils for the compressible Navier–Stokes equations coupled to an additional (LB-based or classical) solver for the energy balance equation. In most cases, the latter relies on a thermal Hermite expansion of the continuous equilibrium distribution function (EDF) to allow for compressibility. Even though recovering the correct equation of state at the Euler level, it has been observed that deviations of local flow temperature from the reference can result in instabilities and/or over-dissipation. The aim of the present study is to evaluate the stability domain of different EDFs, different collision models, with and without the correction terms for the third-order moments. The study is first based on a linear von Neumann analysis. The correction term for the space- and time-discretized equations is derived via a Chapman–Enskog analysis and further corroborated through spectral dispersion–dissipation curves. Finally, a number of numerical simulations are performed to illustrate the proposed theoretical study. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.