  We present the results of an analysis of Wide-field Infrared Survey Explorer
(WISE) observations on the full 2500 deg^2 South Pole Telescope (SPT)-SZ
cluster sample. We describe a process for identifying active galactic nuclei
(AGN) in brightest cluster galaxies (BCGs) based on WISE mid-infrared color and
redshift. Applying this technique to the BCGs of the SPT-SZ sample, we
calculate the AGN-hosting BCG fraction, which is defined as the fraction of
BCGs hosting bright central AGNs over all possible BCGs. Assuming {\bf an
evolving} single-burst stellar population model, we find statistically
significant evidence (>99.9%) for a mid-IR excess at high redshift compared to
low redshift, suggesting that the fraction of AGN-hosting BCGs increases with
redshift over the range of 0 < z < 1.3. The best-fit redshift trend of the
AGN-hosting BCG fraction has the form (1+z)^(4.1+/-1.0). These results are
consistent with previous studies in galaxy clusters as well as field galaxies.
One way to explain this result is that member galaxies at high redshift tend to
have more cold gas. While BCGs in nearby galaxy clusters grow mostly by dry
mergers with cluster members, leading to no increase in AGN activity, BCGs at
high redshift could primarily merge with gas-rich satellites, providing fuel
for feeding AGNs. If this observed increase in AGN activity is linked to
gas-rich mergers, rather than ICM cooling, we would expect to see an increase
in scatter in the P_cav vs L_cool relation at z > 1. Lastly, this work confirms
that the runaway cooling phase, as predicted by the classical cooling flow
model, in the Phoenix cluster is extremely rare and most BCGs have low
(relative to Eddington) black hole accretion rates.
