Aims. INTEGRAL has been monitoring the Galactic center region for more than a decade. Over this time it has detected hundreds of type-I X-ray bursts from the neutron star low-mass X-ray binary 4U 1728–34, also known as the slow burster. Our aim is to study the connection between the persistent X-ray spectra and the X-ray burst spectra in a broad spectral range. Methods. We performed spectral modeling of the persistent emission and the X-ray burst emission of 4U 1728–34 using data from the INTEGRAL JEM-X and IBIS/ISGRI instruments. Results. We constructed a hardness intensity diagram to track spectral state variations. In the soft state, the energy spectra are characterized by two thermal components likely coming from the accretion disc and the boundary/spreading layer, together with a weak hard X-ray tail that we detect in 4U 1728–34 for the first time in the ~40 to 80 keV range. In the hard state, the source is detected up to ~200 keV and the spectrum can be described by a thermal Comptonization model plus an additional component: either a powerlaw tail or reflection. By stacking 123 X-ray bursts in the hard state, we detect emission up to 80 keV during the X-ray bursts. We find that during the bursts the emission above 40 keV decreases by a factor of approximately three with respect to the persistent emission level. Conclusions. Our results suggest that the enhanced X-ray burst emission changes the spectral properties of the accretion disc in the hard state. The likely cause is an X-ray burst induced cooling of the electrons in the inner hot flow near the neutron star.