We present thermal Sunyaev-Zeldovich 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.
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