  We study the nature of feedback mechanisms in the 11 CLASH brightest cluster
galaxies (BCGs) that exhibit extended ultraviolet and nebular line emission
features. We estimate star formation rates (SFRs), dust masses, and starburst
durations using a Bayesian photometry-fitting technique that accounts for both
stellar and dust emission from the UV through far IR. By comparing these
quantities to intracluster medium (ICM) cooling times and free-fall times
derived from X-ray observations and lensing estimates of the cluster mass
distribution, we discover a tight relationship between the BCG SFR and the ICM
cooling time to free-fall time ratio, $t_{cool}/t_{ff}$, with an upper limit on
the intrinsic scatter of 0.15 dex. Furthermore, starburst durations may
correlate with ICM cooling times at a radius of $0.025R_{500}$, and the two
quantities converge upon reaching the Gyr regime. Our results provide a direct
observational link between the thermodynamical state of the ICM and the
intensity and duration of BCG star formation activity, and appear consistent
with a scenario where active galactic nuclei (AGN) induce condensation of
thermally unstable ICM overdensities that fuel long-duration ($>$ 1 Gyr) BCG
starbursts. This scenario can explain (a) how gas with a low cooling time is
depleted without causing a cooling flow and (b) the scaling relationship
between SFR and $t_{cool}/t_{ff}$. We also find that the scaling relation
between SFR and dust mass in BCGs with SFRs $<100$ M$_{\odot}$ yr$^{-1}$ is
similar to star-forming field galaxies; BCGs with large ($>100$ M$_{\odot}$
yr$^{-1}$) SFRs have dust masses comparable to extreme starbursts.
