ive heard powersteering pumps make weird noises especially when it sucks up a bubble (low fluid)
could be belt, could be alternater
somehting else it could be since you say it does it and goes away with temps is a vacuum leak, partially cracked hose can cause a whistle or a whine sometimes, or it could have popped off. this would be my best guess.
its obviously somehting engine driven or engine itself so just check anything that roatates
something driven by the belt. Most likely the alternator.
1991 Toyota Pickup 22R-E 2.4 I4/5 speed
1990 Toyota Cressida 7M-GE 3.0 I6/5-speed manual
mechanic, carpenter, stagehand, rigger, and know-it-all smartass
"You don't get to judge me for how I fix what you break"
It is the alternator with the bearing running out of lubrication it will make noise when the belt is cold and tighter putting more load on the bearing and once it warms up the lube will move around and quiet it down and the belt will stretch a little. The Alternator is the only belt driven device that is geared up to a higher RPM while everything else is slower or about the same speed as the crank. Check for dust around the bearings as well as any play in the shaft of the alternator then you can loosen the belt and try spinning the alternator by hand and see if it feels a little bumpy - that will tell you right away.
A far stretch would be the starter motor gear not returning from the flywheel.