  We present IR and UV photometry for a sample of brightest cluster galaxies
(BCGs). The BCGs are from a heterogeneous but uniformly characterized sample,
the Archive of Chandra Cluster Entropy Profile Tables (ACCEPT), of X-ray galaxy
clusters from the Chandra X-ray telescope archive with published gas
temperature, density, and entropy profiles. We use archival GALEX, Spitzer, and
2MASS observations to assemble spectral energy distributions (SEDs) and colors
for BCGs. We find that while the SEDs of some BCGs follow the expectation of
red, dust-free old stellar populations, many exhibit signatures of recent star
formation in the form of excess UV or mid-IR emission, or both. We establish a
mean near-UV to 2MASS K color of 6.59 \pm 0.34 for quiescent BCGs. We use this
mean color to quantify the UV excess associated with star formation in the
active BCGs. We use fits to a template of an evolved stellar population and
library of starburst models and mid-IR star formation relations to estimate the
obscured star formation rates. Many of the BCGs in X-ray clusters with low
central gas entropy exhibit enhanced UV (38%) and mid-IR emission (43%), above
that expected from an old stellar population. These excesses are consistent
with on-going star formation activity in the BCG, star formation that appears
to be enabled by the presence of high density, X-ray emitting gas in the the
core of the cluster of galaxies. This hot, X-ray emitting gas may provide the
enhanced ambient pressure and some of the fuel to trigger the star formation.
This result is consistent with previous works that showed that BCGs in clusters
with low central gas entropy host H{\alpha} emission-line nebulae and radio
sources, while clusters with high central gas entropy exhibit none of these
features. UV and mid-IR measurements combined provide a complete picture of
unobscured and obscured star formation occurring in these systems.
