We address the cosmological evolution of violent gravitational instability in high-redshift, massive, star-forming galactic discs. To this aim, we integrate in time the equations of mass and energy conservation under self-regulated instability of a two-component disc of gas and stars. The disc is assumed to be continuously fed by cold gas at the average cosmological rate. The gas forms stars and is partly driven away by stellar feedback. The gas and stars flow inward through the disc to a central bulge due to torques that drive angular momentum outwards. The gravitational energy released by the mass inflow down the gravitational potential gradient drives the disc turbulence that maintains the disc unstable with a Toomre instability parameter Q∼ 1, compensating for the dissipative losses of the gas turbulence and raising the stellar velocity dispersion. We follow the velocity dispersion of stars and gas as they heat and cool, respectively, and search for disc ‘stabilization’, to be marked by a low gas velocity dispersion comparable to the speed of sound ∼10 km s−1. We vary the model parameters that characterize the accreted gas fraction, turbulence dissipation rate, star formation rate and stellar feedback. We find that as long as the gas input roughly follows the average cosmological rate, the disc instability is a robust phenomenon at high redshift till z∼ 1, driven by the high surface density and high gas fraction due to the intense cosmological accretion. For a broad range of model parameter values, the discs tend to ‘stabilize’ at z∼ 0–0.5 as they become dominated by hot stars. When the model parameters are pushed to extreme values, the discs may stabilize as early as z∼ 2, with the gas loss by strong outflows serving as the dominant stabilizing factor.