We have obtained the first large sample of accurate temperatures for c
lusters at z > 0.14 from ASCA. We compare the luminosity-temperature (
L-T) distribution for these clusters with the low-redshift sample of D
avid and coworkers and find that there is no evidence for evolution. W
e also find that the intrinsic variance in this relation is roughly co
nstant with redshift. Additionally, there is no detectable change in t
he relationship of optical velocity dispersion to X-ray temperature wi
th redshift. Most cosmological simulations driven primarily by gravity
predict substantial changes in the L-T relation due to the recent rap
id growth of clusters. Our results are consistent either with models i
n which the cluster core entropy is dominated by preheating, or with l
ow-Omega models in which cluster structure does not evolve strongly wi
th time. The intrinsic variance in the L-T relation at a fixed redshif
t can be due to variety of possible causes, e.g., a change in the bary
onic fraction from cluster to cluster, variation in the fraction of th
e total energy in the system arising from shocks or supernova heating,
or variations in the emission measure distributions in multiphase gas
.