We use a suite of hydrodynamical cosmological simulations from the Evolution and Assembly of GaLaxies and their Environments (EAGLE) project to investigate the formation of hot hydrostatic haloes and their dependence on feedback mechanisms. We find that the appearance of a strong bimodality in the probability density function of the ratio of the radiative cooling and dynamical times for halo gas provides a clear signature of the formation of a hot corona. Haloes of total mass 10^11.5–10¹² M_⊙ develop a hot corona independent of redshift, at least in the interval z = 0–4, where the simulation has sufficiently good statistics. We analyse the build-up of the hot gas mass in the halo, M_hot, as a function of halo mass and redshift and find that while more energetic galactic wind powered by SNe increases M_hot, active galactic nucleus feedback reduces it by ejecting gas from the halo. We also study the thermal properties of gas accreting on to haloes and measure the fraction of shock-heated gas as a function of redshift and halo mass. We develop analytic and semi-analytic approaches to estimate a ‘critical halo mass’, M_crit, for hot halo formation. We find that the mass for which the heating rate produced by accretion shocks equals the radiative cooling rate reproduces the mass above which haloes develop a significant hot atmosphere. This yields a mass estimate of M_crit ≈ 10^11.7 M_⊙ at z = 0, which agrees with the simulation results. The value of M_crit depends more strongly on the cooling rate than on any of the feedback parameters.