  We report the results of a study which assembles deep observations with the
ACIS-I instrument on the Chandra Observatory to study the evolution in the core
properties of a sample of galaxy groups and clusters out to redshifts $z\approx
1.3$. A search for extended objects within these fields yields a total of 62
systems for which redshifts are available, and we added a further 24
non-X-ray-selected clusters, to investigate the impact of selection effects and
improve our statistics at high redshift. Six different estimators of cool core
strength are applied to these data: the entropy (K) and cooling time
($t_{cool}$) within the cluster core, the cooling time as a fraction of the age
of the Universe ($t_{cool}/t_{Uni}$), and three estimators based on the
cuspiness of the X-ray surface brightness profile. A variety of statistical
tests are used to quantify evolutionary trends in these cool core indicators.
In agreement with some previous studies, we find that there is significant
evolution in $t_{cool}/t_{Uni}$, but little evolution in $t_{cool}$, suggesting
that gas is accumulating within the core, but that the cooling time deep in the
core is controlled by AGN feedback. We show that this result extends down to
the group regime and appears to be robust against a variety of selection biases
(detection bias, archival biases and biases due to the presence of central
X-ray AGN) which we consider.
