  Restarted radio galaxies represent a unique tool to investigate the duty
cycle of the jet activity in Active Galactic Nuclei. Due to a sharp
discontinuity of the GHz spectral index distribution within its lobes, the
radio galaxy 3C388 has for long being claimed to be a peculiar example of AGN
with multi-epoch activity. In this work we have performed, for the first time,
a spatially resolved study of the radio spectrum of this source down to
MHz-frequencies, aimed at investigating its radiative age and duty cycle. We
have used new low frequency observations at 144 MHz performed with LOFAR and at
350 MHz performed with the VLA combined with archival data at higher
frequencies. We find that the spectral indices in the lower frequency range
144-614 MHz have flatter values ($\sim$0.55-1.14) than those observed in the
higher frequency range 1400-4850 MHz ($\sim$0.75-1.57) but follow the same
distribution across the lobes, with a systematic steepening towards the edges.
However, the spectral shape throughout the source is not uniform and often
deviates from standard models. This suggests that mixing of different particle
populations is occurring, although it remains difficult to understand whether
this is caused by observational limitations (insufficient spatial resolution
and/or projection effects) or by the intrinsic presence of multiple particle
populations, possibly related to the two different outbursts. By using
single-injection radiative models we compute that the total source age is
$\lesssim$80 Myr and that the duty cycle is about $\rm t_{on}/t_{tot}\sim$ 60%,
which is enough to prevent the intracluster medium from cooling according to
X-ray estimates. While to date the radio spectral distribution of 3C388 remains
a rare case among radio galaxies, upcoming multi-frequency surveys will soon
allow us to investigate whether more sources with the same characteristics do
actually exist.
