Prolonged high pitch sound (engine coolant thermo pump?) at startup

Yesterday evening when I started the car there was a high pitch sound similar to the engine coolant thermal bottle pump sound that continued even after the engine starts. (The sound was there from the very beginning of "READY" mode before the engine starts.) I sat in the car for about a minute and couldn't figure out the source of the sound. I did check and make sure it wasn't the radio. Turned off the car then the sound was gone. Started the car again and the sound comes back. Drove a short distance to the exit of the parking lot and came to a full stop, the engine stopped (it usually does), and the sound was gone and never come back since.

This morning I checked all fluid levels OK. The car started without that high pitch sound. Any suggestions/ideas? Thanks!