We analyze the output power versus temperature characteristics of two GaAs/Al0.15Ga0.85As terahertz quantum cascade lasers (THz-QCLs) with maximum operating temperature $T_{max}=200$ and 177 K as well as of one GaAs/Al0.30Ga0.70As THz-QCL with $T_{max}=150$ K and identify the thermally-activated leakage paths responsible for the laser performance degradation as the temperature increases. We identify the specific carrier leakage path active in each THz-QCL structure and are able to reconstruct the output power versus temperature profile over the entire laser operation range. We find that using high barriers in the active region design virtually eliminates carrier leakage from the upper laser level into the continuum, opening a non-radiative scattering path from the upper into the lower laser level parallel to standard electron-LO-phonon emission. This effect, together with the reduced leakage from the lower laser level into the continuum in the high-barrier device, significantly contributes to the $T_{max}$ decrease from 177 to 150 K. We further show how electron leakage from the lower laser level into the continuum is enhanced in a GaAs/Al0.15Ga0.85As design with thin barriers, significantly improving the laser performance ( $T_{max}=200$ K). Finally, we propose future design strategies for highly temperature-insensitive THz-QCLs. Our approach offers a straightforward method to analyze and troubleshoot thermally-activated carrier leakage dynamics in THz-QCLs.