  We present thermal Sunyaev-Zel'dovich effect (SZE) measurements for 42 galaxy
clusters observed at 150 GHz with the APEX-SZ experiment. For each cluster, we
model the pressure profile and calculate the integrated Comptonization $Y$ to
estimate the total thermal energy of the intracluster medium (ICM). We compare
the measured $Y$ values to X-ray observables of the ICM from the literature
(cluster gas mass $M_{\rm{gas}}$, temperature $T_X$, and $Y_X
=M_{\rm{gas}}T_X$) that relate to total cluster mass. We measure power law
scaling relations, including an intrinsic scatter, between the SZE and X-ray
observables for three subsamples within the set of 42 clusters that have
uniform X-ray analysis in the literature. We observe that differences between
these X-ray analyses introduce significant variability into the measured
scaling relations, particularly affecting the normalization. For all three
subsamples, we find results consistent with a self-similar model of cluster
evolution dominated by gravitational effects. Comparing to predictions from
numerical simulations, these scaling relations prefer models that include
cooling and feedback in the ICM. Lastly, we measure an intrinsic scatter of
$\sim28$ per cent in the $Y-Y_X\,$ scaling relation for all three subsamples.
