We measured the gain dynamics at the ground-state transition in an electrically pumped InP/AlGaInP quantum-dot optical amplifier at room temperature by femtosecond differential transmission. The gain shows an ultrafast recovery within 200 fs, even faster than in state-of-the-art InAs/GaAs quantum-dot amplifiers. This finding, likely to be due to the less confined and more closely spaced hole levels in InP dots, is promising for optical signal processing at high bit rates. We furthermore measured the pump-induced refractive index changes and deduced a linewidth enhancement factor similar to the one in InAs/GaAs quantum dots.