We present the first dedicated X-ray study of the supernova remnant (SNR) G32.8-0.1 (Kes 78) with Suzaku. X-ray emission from the whole SNR shell has been detected for the first time. The X-ray morphology is well correlated with the emission from the radio shell, while anti-correlated with the molecular cloud found in the SNR field. The X-ray spectrum shows not only conventional low-temperature (kT ~ 0.6 keV) thermal emission in a non-equilibrium ionization state, but also a very high temperature (kT ~ 3.4 keV) component with a very low ionization timescale (~ 2.7e9 cm^{-3}s), or a hard non-thermal component with a photon index Gamma~2.3. The average density of the low-temperature plasma is rather low, of the order of 10^{-3}--10^{-2} cm^{-3}, implying that this SNR is expanding into a low-density cavity. We discuss the X-ray emission of the SNR, also detected in TeV with H.E.S.S., together with multi-wavelength studies of the remnant and other gamma-ray emitting SNRs, such as W28 and RCW 86. Analysis of a time-variable source, 2XMM J185114.3-000004, found in the northern part of the SNR, is also reported for the first time. Rapid time variability and a heavily absorbed hard X-ray spectrum suggest that this source could be a new supergiant fast X-ray transient.
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