The effect of elevated temperatures on the optical and structural stability of MOSi2/Si and Mo/C/Si/C multilayer coatings was investigated. The multilayer mirrors were designed for normal-incidence reflectivity at a wavelength of about 13.5 nm. The multilayers were deposited by dc-magnetron sputtering and subsequently annealed at temperatures of 400 degrees C and 500 degrees C for 1, 10 and 100 hours. X-ray scattering, transmission electron microscopy, atomic force microscopy and normal-incidence reflectivity measurements were used for the characterization of the multilayer structures. We achieved maximal normal-incidence reflectivities of 41.2 % and 59.6 % for as-deposited MOSi2/S and Mo/C/Si/C multilayer mirrors. While the optical properties of Mo/C/Si/C multilayers changed monotonically during annealing time at temperatures of more than 400 degrees C, the MOSi2/Si mulitilayers showed a superior thermal stability up to 500 degrees C. New barrier layer materials were also suggested to enhance the thermal stability of Mo/Si multilayers. Interface-engineered Mo/Si multilayer mirrors were designed to combine both a high reflectivity of more than 60 % at 13.5 nm and a superior long-term thermal stability of up to 500 degrees C.