  We use the EAGLE cosmological simulations to study the evolution of the
vertical velocity dispersion of cold gas, $\sigma_{z}$, in central disc
galaxies and its connection to stellar feedback, gravitational instabilities,
cosmological gas accretion and galaxy mergers. To isolate the impact of
feedback, we analyse runs that turn off stellar and (or) AGN feedback in
addition to a run that includes both. The evolution of $\sigma_z$ and its
dependence on stellar mass and star formation rate in EAGLE are in good
agreement with observations. Galaxies hosted by haloes of similar virial mass,
$\rm M_{200}$, have similar $\sigma_z$ values even in runs where feedback is
absent. The prevalence of local instabilities in discs is uncorrelated with
$\sigma_z$ at low redshift and becomes only weakly correlated at high redshifts
and in galaxies hosted by massive haloes. $\sigma_z$ correlates most strongly
with the specific gas accretion rate onto the disc as well as with the degree
of misalignment between the inflowing gas and the disc's rotation axis. These
correlations are significant across all redshifts and halo masses, with
misaligned accretion being the primary driver of high gas turbulence at
redshifts $z \lesssim 1$ and for halo masses $\rm M_{200} \lesssim 10^{11.5}
M_{\odot}$. Galaxy mergers increase $\sigma_z$, but because they are rare in
our sample, they play only a minor role in its evolution. Our results suggest
that the turbulence of cold gas in EAGLE discs results from a complex interplay
of different physical processes whose relative importance depends on halo mass
and redshift.
