We perform a detailed analysis of the thermal structure of the region above the post-eruption arcade for a flare that occurred on 2011 October 22. During this event, a sheet of hot plasma is visible above the flare loops in the 131 Å bandpass of the Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory. Supra-arcade downflows (SADs) are observed traveling sunward through the post-eruption plasma sheet. We calculate differential emission measures using the AIA data and derive an emission measure weighted average temperature in the supra-arcade region. In areas where many SADs occur, the temperature of the supra-arcade plasma tends to increase, while in areas where no SADs are observed, the temperature tends to decrease. We calculate the plane-of-sky velocities in the supra-arcade plasma and use them to determine the potential heating due to adiabatic compression and viscous heating. Of the 13 SADs studied, 10 have noticeable signatures in both the adiabatic and the viscous terms. The adiabatic heating due to compression of plasma in front of the SADs is on the order of 0.1-0.2 MK/s, which is similar in magnitude to the estimated conductive cooling rate. This result supports the notion that SADs contribute locally to the heating of plasma in the supra-arcade region. We also find that in the region without SADs, the plasma cools at a rate that is slower than the estimated conductive cooling, indicating that additional heating mechanisms may act globally to keep the plasma temperature high.
