Does the pump just fails at once? Does it relate to the ac unit? I'm having issues with the ac not working properly at low speeds and/or traffic lights, the other day temp light came on yellow and turned red as I was getting on the ramp to the expressway. Once I got some speed light went out and ac started to work normally.
Could be the electric fan behind the radiator is not operating correctly, and once you have the car up to speed to enough airflow through the radiator, both the A/C and cooling is operating (without the fan.) I'd check that the fan is running at significant speed (it's variable speed) when stopped with the A/C running. I can hear the fan spool up when I'm stopped and switch on the A/C.