  We present a detailed investigation of the X-ray luminosity (Lx)-gas
temperature (Tvir) relation of the complete X-ray flux-limited sample of the 64
brightest galaxy clusters in the sky (HIFLUGCS). We study the influence of two
astrophysical processes, active galactic nuclei (AGN) heating and intracluster
medium (ICM) cooling, on the Lx-Tvir relation, simultaneously for the first
time. We determine best-fit relations for different subsamples using the
cool-core strength and the presence of central radio activity as selection
criteria. We find the strong cool-core clusters (SCCs) with short cooling times
(< 1Gyr)to display the steepest relation (Lx ~ Tvir^{3.33}) and the
non-cool-core clusters (NCCs) with long cooling times (> 7.7Gyr) to display the
shallowest (Lx ~ Tvir^{2.42}). This has the simple implication that on the
high-mass scale (Tvir > 2.5keV) the steepening of the Lx-Tvir relation is
mainly due to the cooling of the intracluster medium gas. We propose that ICM
cooling and AGN heating are both important in shaping the Lx-Tvir relation but
on different length-scales. While our study indicates that ICM cooling
dominates on cluster scales (Tvir > 2.5keV), we speculate that AGN heating
dominates the scaling relation in poor clusters and groups (Tvir < 2.5keV). The
intrinsic scatter about the Lx-Tvir relation in X-ray luminosity for the whole
sample is 45.4% and varies from a minimum of 34.8% for weak cool-core clusters
to a maximum of 59.4% for clusters with no central radio source. We find that
after excising the cooling region, the scatter in the Lx-Tvir relation drops
from 45.4% to 39.1%, implying that the cooling region contributes ~ 27% to the
overall scatter. Lastly, we find the true SCC fraction to be 25% lower than the
observed one and the true normalizations of the Lx-Tvir relations to be lower
by 12%, 7%, and 17% for SCC, WCC, and NCC clusters, respectively. [abridged]
