The key task in the solidification of high-level waste (HLW) into a magnesium potassium phosphate (MPP) compound is the immobilization of mobile cesium isotopes, the activity of which provides the main contribution to the total HLW activity. In addition, the obtained compound containing heat-generating radionuclides can be significantly heated, which increases the necessity of its thermal stability. The current work is aimed at assessing the impact of various methodological approaches to HLW solidification on the thermal stability of the MPP compound, which is evaluated by the mechanical strength of the compound and its resistance to cesium leaching. High-salt surrogate HLW solution (S-HLW) used in the investigation was prepared for solidification by adding sorbents of various types binding at least 93% of 137Cs: ferrocyanide K-Ni (FKN), natural zeolite (NZ), synthetic zeolite Na-mordenite (MOR), and silicotungstic acid (STA). Prepared S-HLW was solidified into the MPP compound. Wollastonite (W) and NZ as fillers were added to the compound composition in the case of using FKN and STA, respectively. It was found that heat treatment up to 450 °C of the compound containing FKN and W (MPP-FKN-W) almost did not affect its compressive strength (about 12–19 МPa), and it led to a decrease of high compressive strength (40–50 MPa) of the compounds containing NZ, MOR, and STA (MPP-NZ, MPP-MOR, and MPP-STA-NZ, respectively) by an average of 2–3 times. It was shown that the differential leaching rate of 137Cs on the 28th day from MPP-FKN-W after heating to 250 °C was 5.3 × 10−6 g/(cm2∙day), however, at a higher temperature, it increased by 20 and more times. The differential leaching rate of 137Cs from MPP-NZ, MPP-MOR, and MPP-STA-NZ had values of (2.9–11) × 10−5 g/(cm2∙day), while the dependence on the heat treatment temperature of the compound was negligible.