Surface temperature is among crucial factors which control wear during sliding dry contact. Using computer modeling, we study the possibility to achieve close to zero rate of surface wear during sliding friction of the special type of materials which possess negative thermal expansion. The numerical simulations reveal two wear regimes for materials with negative thermal expansion coefficient as dependent on the applied normal stress level. When the applied stress is lower than that of a critical level, a steady almost zero wear rate and nanorough surface are achieved during friction. Otherwise, wear rate is of the same order of magnitude as for "traditional" materials with positive thermal expansion coefficient. The critical stress value is analyzed depending on the material's mechanical, thermophysical, and surface roughness characteristics.