Radiative cooling may plausibly cause hot gas in the centre of a massive galaxy, or galaxy cluster, to become gravitationally unstable. The subsequent collapse of this gas on a dynamical time-scale can provide an abundant source of fuel for active galactic nucleus (AGN) heating and star formation. Thus, this mechanism provides a way to link the AGN accretion rate to the global properties of an ambient cooling flow, but without the implicit assumption that the accreted material must have flowed on to the black hole from tens of kpc away. It is shown that a fuelling mechanism of this sort naturally leads to a close balance between AGN heating and the radiative cooling rate of the hot, X-ray-emitting halo. Furthermore, AGN powered by cooling-induced gravitational instability would exhibit characteristic duty cycles (δ) which are redolent of recent observational findings: δ∝ LX/σ3*, where LX is the X-ray luminosity of the hot atmosphere and σ* is the central stellar velocity dispersion of the host galaxy. Combining this result with well-known scaling relations, we deduce a duty cycle for radio AGNs in elliptical galaxies that is approximately ∝ M1.5BH, where MBH is the central black hole mass. Outburst durations and Eddington ratios are also given. Based on the results of this study, we conclude that gravitational instability could provide an important mechanism for supplying fuel to AGNs in massive galaxies and clusters, and warrants further investigation.