(Abridged) We present the results of an X-ray analysis of 80 galaxy clusters selected in the 2500 deg^2 South Pole Telescope survey and observed with the Chandra X-ray Observatory. We divide the full sample into subsamples of ~20 clusters based on redshift and central density, performing an X-ray fit to all clusters in a subsample simultaneously, assuming self-similarity of the temperature profile. This approach allows us to constrain the shape of the temperature profile over 0<r<1.5R500, which would be impossible on a per-cluster basis, since the observations of individual clusters have, on average, 2000 X-ray counts. The results presented here represent the first constraints on the evolution of the average temperature profile from z=0 to z=1.2. We find that high-z (0.6<z<1.2) clusters are slightly (~40%) cooler both in the inner (r<0.1R500) and outer (r>R500) regions than their low-z (0.3<z<0.6) counterparts. Combining the average temperature profile with measured gas density profiles from our earlier work, we infer the average pressure and entropy profiles for each subsample. Overall, our observed pressure profiles agree well with earlier lower-redshift measurements, suggesting minimal redshift evolution in the pressure profile outside of the core. We find no measurable redshift evolution in the entropy profile at r<0.7R500. We observe a slight flattening of the entropy profile at r>R500 in our high-z subsample. This flattening is consistent with a temperature bias due to the enhanced (~3x) rate at which group-mass (~2 keV) halos, which would go undetected at our survey depth, are accreting onto the cluster at z~1. This work demonstrates a powerful method for inferring spatially-resolved cluster properties in the case where individual cluster signal-to-noise is low, but the number of observed clusters is high.
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