Spin Hall nano-oscillators (SHNOs) exploiting current-driven magnetization auto-oscillation have recently received much attention because of their potential for oscillator-based neuromorphic computing. Widespread neuromorphic application with SHNOs requires an energy-efficient way to tune oscillation frequency in broad ranges and store trained frequencies in SHNOs without the need for additional memory circuitry. Voltage control of oscillation frequency of SHNOs was experimentally demonstrated, but the voltage-driven frequency tuning was volatile and limited to megahertz ranges. Here, we show that the frequency of SHNO is controlled up to 2.1 GHz by a moderate electric field of 1.25 MV/cm. The large frequency tuning is attributed to the voltage-controlled magnetic anisotropy (VCMA) in a perpendicularly magnetized Ta/Pt/[Co/Ni]n/Co/AlOx structure. Moreover, non-volatile VCMA effect enables control of the cumulative frequency using repetitive voltage pulses, which mimic the potentiation and depression functions of biological synapses. Our results suggest that the voltage-driven frequency tuning of SHNOs facilitates the development of energy-efficient spin-based neuromorphic devices.