Abstract Recently, slowly cooling white dwarfs (WDs) are a new class of WD that have been identified in two globular clusters (namely M13 and NGC 6752), showing a horizontal branch (HB) morphology with an extended blue tail. The cooling rate of these WDs is reduced by stable thermonuclear hydrogen burning in their residual envelope, and they are thought to originate by stars that populate the blue tail of the HB and then skip the asymptotic giant branch phase. Consistently, no evidence of such kind of WDs has been found in M3, a similar cluster with no blue extension of the HB. To further explore this phenomenon, we took advantage of deep photometric data acquired with the Hubble Space Telescope in the near-ultraviolet and investigated the bright portion of the WD cooling sequence in M5, another Galactic globular cluster with HB morphology similar to M3. The normalized WD luminosity function derived in M5 was found to be impressively similar to that observed in M3, in agreement with the fact that the stellar mass distribution along the HB of these two systems is almost identical. The comparison with theoretical predictions is consistent with the fact that the cooling sequence in this cluster is populated by canonical (fast cooling) WDs. Thus, the results presented in this paper provide further support to the scenario proposing a direct causal connection between the slow cooling WD phenomenon and the horizontal branch morphology of the host stellar cluster.