We designed and demonstrated a terahertz quantum cascade laser based on indirect pump injection to the upper lasing state and phonon scattering extraction from the lower lasing state. By employing a rate equation formalism and a genetic algorithm, an optimized active region design with four-well GaAs/Al0.25Ga0.75As cascade module was obtained and epitaxially grown. A figure of merit which is defined as the ratio of modal gain versus injection current was maximized at 150 K. A fabricated device with a Au metal-metal waveguide and a top n+ GaAs contact layer lased at 2.4 THz up to 128.5 K, while another one without the top n+ GaAs lased up to 152.5 K (1.3ħω/kB). The experimental results have been analyzed with rate equation and nonequilibrium Green's function models. A high population inversion is achieved at high temperature using a small oscillator strength of 0.28, while its combination with the low injection coupling strength of 0.85 meV results in a low current. The carefully engineered wavefunctions enhance the quantum efficiency of the device and therefore improve the output optical power even with an unusually low injection coupling strength.