We present the 0.6 < z < 2.6 evolution of the ionized gas velocity dispersion in 175 star-forming disk galaxies based on data from the full KMOS ^ { 3 D } integral field spectroscopic survey . In a forward-modelling Bayesian framework including instrumental effects and beam-smearing , we fit simultaneously the observed galaxy velocity and velocity dispersion along the kinematic major axis to derive the intrinsic velocity dispersion \sigma _ { 0 } . We find a reduction of the average intrinsic velocity dispersion of disk galaxies as a function of cosmic time , from \sigma _ { 0 } \sim 45  km s ^ { -1 } at z \sim 2.3 to \sigma _ { 0 } \sim 30  km s ^ { -1 } at z \sim 0.9 . There is substantial intrinsic scatter ( \sigma _ { \sigma _ { 0 } , { int } } \approx 10  km s ^ { -1 } ) around the best-fit \sigma _ { 0 } - z -relation beyond what can be accounted for from the typical measurement uncertainties ( \delta \sigma _ { 0 } \approx 12  km s ^ { -1 } ) , independent of other identifiable galaxy parameters . This potentially suggests a dynamic mechanism such as minor mergers or variation in accretion being responsible for the scatter . Putting our data into the broader literature context , we find that ionized and atomic+molecular velocity dispersions evolve similarly with redshift , with the ionized gas dispersion being \sim 10 - 15  km s ^ { -1 } higher on average . We investigate the physical driver of the on average elevated velocity dispersions at higher redshift , and find that our galaxies are at most marginally Toomre-stable , suggesting that their turbulent velocities are powered by gravitational instabilities , while stellar feedback as a driver alone is insufficient . This picture is supported through comparison with a state-of-the-art analytical model of galaxy evolution .