Type I X-ray bursts in GS 1826-24, and in several other systems, may induce cooling of the hot inner accretion flow that surrounds the bursting neutron star. Given that GS 1826-24 remained persistently in the hard state over the period 2003-2008 and presented regular bursting properties, we stacked the spectra of the X-ray bursts detected by INTEGRAL (JEM-X and ISGRI) and XMM-Newton (RGS) during that period to study the effect of the burst photons on the properties of the Comptonizing medium. The extended energy range provided by these instruments allows the simultaneous observation of the burst and persistent emission spectra. We detect an overall change in the shape of the persistent emission spectrum in response to the burst photon shower. For the first time, we observe simultaneously a drop in the hard X-ray emission, together with a soft X-ray excess with respect to the burst blackbody emission. The hard X-ray drop can be explained by burst-induced coronal cooling, while the bulk of the soft X-ray excess can be described by fitting the burst emission with an atmosphere model, instead of a simple blackbody model. Traditionally, the persistent emission was assumed to be invariant during X-ray bursts, and more recently to change only in normalization but not in spectral shape; the observed change in the persistent emission level during X-ray bursts may thus trigger the revision of existing neutron star mass-radius constraints, as the derived values rely on the assumption that the persistent emission does not change during X-ray bursts. The traditional burst fitting technique leads to up to a 10% overestimation of the bolometric burst flux in GS 1826-24, which significantly hampers the comparisons of the KEPLER and MESA model against this textbook burster.