Understanding of damping processes in ferromagnetic thin films at elevated temperatures has significant implications for heat-assisted magnetic recording, spin-transfer torque memory, and magnetic sensors operating at high temperatures. Through cavity-based high-temperature ferromagnetic resonance (FMR) measurements, this work examined the FMR linewidth and damping properties of continuous cubic FePt thin films at elevated temperatures. The data show that the FMR linewidth and the Gilbert damping constant both increase monotonically when temperature is increased from room temperature toward the Curie temperature. This temperature dependence is opposite to that observed previously in FePt thin films that are granular, rather than continuous, and have L10 structure, rather than cubic structure; in those films, the FMR linewidth decreases monotonically with an increase in temperature [PR Applied 10, 054046 (2018)]. These opposite results originate from the difference in the crystalline structure and microstructure of the films. In the previous work, the granular L10-order FePt films hold dense material imperfection and thereby may host strong two-magnon scattering (TMS); the TMS-produced damping decreases with an increase in temperature, giving rise to reduced FMR linewidths at high temperatures. In the current work, the continuous cubic FePt films have much less imperfection and thereby host weak TMS, and the dominant damping mechanism is spin-flip magnon-electron scattering (SF-MES). The SF-MES process becomes stronger with an increase in temperature, giving rise to larger linewidth and higher damping at high temperatures. This work and the previous work together demonstrate that for a given thin-film material, the temperature dependence of the FMR linewidth critically relies on the structural properties of the film. They also indicate that one can engineer damping in magnetic thin films through the control of the structural properties of the films.