ASCA and ROSAT X-ray data are used to obtain two-dimensional maps of the gas temperature, pressure, and specific entropy in the Triangulum Australis cluster of galaxies. We find that this hot (T-e = 10.3 +/- 0.8 keV) system probably has a temperature peak (T-e > 12 keV) at the cluster core, which approximately corresponds to the adiabatic relation. An underdense gas sector, found in the ROSAT cluster image eastward of the core, has a higher temperature than average at that radius. At this higher temperature, the gas pressure in this sector is equal to that of the rest of the cluster at the same radius, but the specific entropy of this gas is significantly higher (although the temperature difference itself is only marginally significant). We speculate that the existence of this region of underdense high-entropy gas, as well as the adiabatic central temperature peak, indicate recent or ongoing heating of the intergalactic medium in this cluster. The most probable source of such heating is a subcluster merger, for which the hydrodynamic simulations predict a qualitatively similar temperature structure. We point out that entropy maps can provide a physically meaningful way of diagnosing merging clusters and comparing the predictions of merger simulations to the data.