We report 78 secondary eclipse depths for a sample of 36 transiting hot Jupiters observed at 3.6 and 4.5 μm using the Spitzer Space Telescope. Our eclipse results for 27 of these planets are new, and include highly irradiated worlds such as KELT-7b, WASP-87b, WASP-76b, and WASP-64b, and important targets for James Webb Space Telescope such as WASP-62b. We find that WASP-62b has a slightly eccentric orbit ( $e\cos \omega =0.00614\pm 0.00064 ), and we confirm the eccentricity of HAT-P-13b and WASP-14b. The remainder are individually consistent with circular orbits, but we find statistical evidence for eccentricity increasing with orbital period in our range from 1 to 5 days. Our day-side brightness temperatures for the planets yield information on albedo and heat redistribution, following Cowan & Agol (2011). Planets having maximum day-side temperatures exceeding ˜2200 K are consistent with having zero albedo and a distribution of stellar irradiance uniformly over the day-side hemisphere. Our most intriguing result is that we detect a systematic difference between the emergent spectra of these hot Jupiters as compared to blackbodies. The ratio of observed brightness temperatures, Tb(4.5)/Tb(3.6), increases with equilibrium temperature by 100 ± 24 parts-per-million per Kelvin, over the entire temperature range in our sample (800-2500 K). No existing model predicts this trend over such a large range of temperature. We suggest that this may be due to a structural difference in the atmospheric temperature profiles of real planetary atmospheres as compared to models.