The detailed knowledge of plasma heating and acceleration region properties presents a major observational challenge in solar flare physics. Using the Ramaty High Energy Solar Spectroscopic Imager (RHESSI), the high temperature differential emission measure, DEM(T), and the energy-dependent spatial structure of solar flare coronal sources were studied quantitatively. The altitude of the coronal X-ray source was observed to increase with energy by ∼+0.2 arcsec/keV between 10 and 25 keV. Although an isothermal model can fit the thermal X-ray spectrum observed by RHESSI, such a model cannot account for the changes in altitude, and multi-thermal coronal sources are required where the temperature increases with altitude. For the first time, we show how RHESSI imaging information can be used to constrain the DEM(T) of a flaring plasma. We developed a thermal bremsstrahlung X-ray emission model with inhomogeneous temperature and density distributions to simultaneously reproduce i) DEM(T); ii) altitude as a function of energy; and iii) vertical extent of the flaring coronal source versus energy. We find that the temperature-altitude gradient in the region is ∼+0.08 keV/arcsec (∼1.3 MK/Mm). Similar altitude-energy trends in other flares suggest that the majority of coronal X-ray sources are multi-thermal and have strong vertical temperature and density gradients with a broad DEM(T).