  We present a study of 107 galaxies, groups, and clusters spanning ~3 orders
of magnitude in mass, ~5 orders of magnitude in central galaxy star formation
rate (SFR), ~4 orders of magnitude in the classical cooling rate (dM/dt) of the
intracluster medium (ICM), and ~5 orders of magnitude in the central black hole
accretion rate. For each system in this sample, we measure dM/dt using archival
Chandra X-ray data and acquire the SFR and systematic uncertainty in the SFR by
combining over 330 estimates from dozens of literature sources. With these
data, we estimate the efficiency with which the ICM cools and forms stars,
finding e_cool = SFR/(dM/dt) = 1.4 +/- 0.4% for systems with dM/dt > 30
Msun/yr. For these systems, we measure a slope in the SFR-dM/dt relation
greater than unity, suggesting that the systems with the strongest cool cores
are also cooling more efficiently. We propose that this may be related to, on
average, higher black hole accretion rates in the strongest cool cores, which
could influence the total amount (saturating near the Eddington rate) and
dominant mode (mechanical vs radiative) of feedback. For systems with dM/dt <
30 Msun/yr, we find that the SFR and dM/dt are uncorrelated, and show that this
is consistent with star formation being fueled at a low (but dominant) level by
recycled ISM gas in these systems. We find an intrinsic log-normal scatter in
SFR at fixed dM/dt of 0.52 +/- 0.06 dex, suggesting that cooling is tightly
self-regulated over very long timescales, but can vary dramatically on short
timescales. There is weak evidence that this scatter may be related to the
feedback mechanism, with the scatter being minimized (~0.4 dex) in systems for
which the mechanical feedback power is within a factor of two of the cooling
luminosity.
