Observed galaxies with high stellar masses or in dense environments have low specific star formation rates, i.e. they are quenched. Based on cosmological hydrodynamic simulations that include a prescription where quenching occurs in regions dominated by hot (>105.4 K) gas, we argue that this hot gas quenching in haloes >1012 M⊙ drives both mass quenching (i.e. central quenching) and environment quenching (i.e. satellite quenching). These simulations reproduce a broad range of locally observed trends among quenching, halo mass, stellar mass, environment, and distance to halo centre. Mass quenching is independent of environment because ∼1012–1013 M⊙ ‘mass quenching haloes’ inhabit a large range of environments. On the other hand, environment quenching is independent of stellar mass because galaxies of all stellar masses may live in dense environments as satellites of groups and clusters. As in observations, the quenched fraction of satellites increases with halo mass and decreases with distance to the centre of the group or cluster. We investigate pre-processing in group haloes, ejected former satellites, and hot gas that extends beyond the virial radius. The agreement of our model with key observational trends suggests that hot gas in massive haloes plays a leading role in quenching low-redshift galaxies.