The R-R Pegasus used distilled water injection at the combustor to keep turbine inlet temps within acceptable limits during hover. The injected water reduces gas temps thru latent heat of evaporation. Since the injected water is inert, it does not contribute to the combustion process, but it does absorb heat which reduces the gas temp at the start of combustion, and thus allows a greater mass of fuel to be added to the existing airflow without exceeding turbine inlet temp limits.
Technically, the water injection does not "cool the engine". It simply lowers the temperature of the compressed air flow at the start of combustion. Gas turbine engines are typically limited by turbine inlet temps, or basically peak combustion cycle temps. So reducing gas temp at the start of combustion allows for a larger temperature rise during combustion. This means more fuel can be burned, and more heat energy added to the exhaust gas.