Operating temp and ac fluctuating

Hello,
My first post to the group, for what that is worth.
My alternator belt broke on my 2001 Elantra. I replaced the pump (which needed to be replaced) as well as the belts. After the belt
broke I limped along for a few miles which may provide clues as well.
After installing the new pump the car now has fluctuating operating temperature. It is normal when moving but rises when idling.
Additionally the A/C blows cold when I first start operating the car but fairly quickly stops cooling.
I suspect that the broken belt may have caused some secondary damage. I want to get in with a flashlight and do a visual inspection after the sun goes down this evening, but I am seeking advice as to the likely culprits in this scenario.
Perhaps it is just a coincidence that these new problem arose when they did, but it seems unlikely.
Thanks.
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As far as the overheating while idling - make sure that the fan kicks on when it starts to heat up. Insure that the fan is connected (assuming it is electrical)
Steve Harder-Kucera wrote:

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While checking the belts I considered the obvious and examined the coolant level. It was very low. Filled up the radiator and both problems went away.
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Is the belt properly tightened? That would be the first thing I'd check. It may be slipping and not turning the AC compressor enough at higher rpm. Could be a defective water pump also that is causing belt slip. I wonder if the thermostat was damaged when you drove the car hot and now it dos not open/close properly especially at low flow rates at idle.
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