This paper presents the first measurement of the radio luminosity function of 'jet-mode' (radiatively inefficient) radio-AGN out to z=1, in order to investigate the cosmic evolution of radio-AGN feedback. Eight radio source samples are combined to produce a catalogue of 211 radio-loud AGN with 0.5<z<1.0, which are spectroscopically classified into jet-mode and radiative-mode (radiatively efficient) AGN classes. Comparing with large samples of local radio-AGN from the Sloan Digital Sky Survey, the cosmic evolution of the radio luminosity function of each radio-AGN class is independently derived. Radiative-mode radio-AGN show an order of magnitude increase in space density out to z~1 at all luminosities, consistent with these AGN being fuelled by cold gas. In contrast, the space density of jet-mode radio-AGN decreases with increasing redshift at low radio luminosities (L_1.4GHz_<~10^24^W/Hz) but increases at higher radio luminosities. Simple models are developed to explain the observed evolution. In the best-fitting models, the characteristic space density of jet-mode AGN declines with redshift in accordance with the declining space density of massive quiescent galaxies, which fuel them via cooling of gas in their hot haloes. A time delay of 1.5-2Gyr may be present between the quenching of star formation and the onset of jet-mode radio-AGN activity. The behaviour at higher radio luminosities can be explained either by an increasing characteristic luminosity of jet-mode radio-AGN activity with redshift (roughly as (1+z)^3^) or if the jet-mode radio-AGN population also includes some contribution of cold-gas-fuelled sources seen at a time when their accretion rate was low. Higher redshifts measurements would distinguish between these possibilities.