  We present a multi-wavelength study of 90 brightest cluster galaxies (BCGs)
in a sample of galaxy clusters selected via the Sunyaev Zel'dovich effect by
the South Pole Telescope, utilizing data from various ground- and space-based
facilities. We infer the star formation rate (SFR) for the BCG in each cluster,
based on the UV and IR continuum luminosity, as well as the [O II] emission
line luminosity in cases where spectroscopy is available, finding 7 systems
with SFR > 100 Msun/yr. We find that the BCG SFR exceeds 10 Msun/yr in 31 of 90
(34%) cases at 0.25 < z < 1.25, compared to ~1-5% at z ~ 0 from the literature.
At z > 1, this fraction increases to 92(+6)(-31)%, implying a steady decrease
in the BCG SFR over the past ~9 Gyr. At low-z, we find that the specific star
formation rate in BCGs is declining more slowly with time than for field or
cluster galaxies, most likely due to the replenishing fuel from the cooling ICM
in relaxed, cool core clusters. At z > 0.6, the correlation between cluster
central entropy and BCG star formation - which is well established at z ~ 0 -
is not present. Instead, we find that the most star-forming BCGs at high-z are
found in the cores of dynamically unrelaxed clusters. We investigate the
rest-frame near-UV morphology of a subsample of the most star-forming BCGs
using data from the Hubble Space Telescope, finding complex, highly asymmetric
UV morphologies on scales as large as ~50-60 kpc. The high fraction of
star-forming BCGs hosted in unrelaxed, non-cool core clusters at early times
suggests that the dominant mode of fueling star formation in BCGs may have
recently transitioned from galaxy-galaxy interactions to ICM cooling.
