We present a detailed X-ray study of the intracluster medium (ICM) of the nearby, cool-core galaxy cluster Abell 478, with Chandra and XMM observations. Using a wavelet smoothing hardness analysis, we derive detailed temperature maps of A478, revealing a surprising amount of temperature structure. The broad band Chandra spectral fits yield temperatures which are significantly hotter than those from XMM, but the Fe ionization temperature shows good agreement. We show that the temperature discrepancy is slightly reduced when comparing spectra from regions selected to enclose nearly isothermal gas. However, by simulating multi-temperature spectra and fitting them with a single temperature model, we find no significant difference between Chandra and XMM, indicating that non-isothermality cannot fully explain the discrepancy. We have discovered 4 hot spots located between 30--50 kpc from the cluster center, where the gas temperature is roughly a factor of 2 higher than in the surrounding material. We estimate the combined excess thermal energy present in these hot spots to be (3+/-1)x10^59 erg. The location of and amount of excess energy present in the hot spots are suggestive of a common origin within the cluster core, which hosts an active galactic nucleus. This cluster also possesses a pair of X-ray cavities coincident with weak radio lobes, as reported in a previous analysis, with an associated energy <10% of the thermal excess in the hot spots. The presence of these hot spots could indicate strong-shock heating of the ICM from the central radio source -- one of the first such detections in a cool core cluster. We also probe the mass distribution in the core and find it to be characterized by a logarithmic slope of -0.35+/-0.22, which is significantly flatter than an NFW cusp of -1. (abridged)
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