We investigate the thermonuclear bursting behaviour of IGR J17473−2721, an X-ray transient that in 2008 underwent a 6-month long outburst, starting (unusually) with an X-ray burst. We detected a total of 57 thermonuclear bursts throughout the outburst with AGILE, Swift, Rossi X-ray Timing Explorer (RXTE) and the INTErnational Gamma-Ray Astrophysics Laboratory(INTEGRAL). The wide range of inferred accretion rates (between <1 and ≃20 per cent of the Eddington accretion rate ) spanned during the outburst allows us to study changes in the nuclear burning processes and to identify up to seven different phases. The burst rate increased gradually with the accretion rate until it dropped (at a persistent flux corresponding to ≃15 per cent of ) a few days before the outburst peak, after which bursts were not detected for a month. As the persistent emission subsequently decreased, the bursting activity resumed at a much lower rate than during the outburst rise. This hysteresis may arise from the thermal effect of the accretion on the surface nuclear burning processes, and the time-scale is roughly consistent with that expected for the neutron star crust thermal response. On the other hand, an undetected ‘superburst’, occurring within a data gap near the outburst peak, could have produced a similar quenching of burst activity.