I have a 2007 jeep liberty with a belt that slips under unusual conditions. It doesn't slip on start up, but it slips almost every time about 10-20 seconds after and it will slip for a few seconds. The only things that have been replaced are the belt, tensioner, and alternator. The only things that seem to affect it are temperature, and using the a/c. I have used the a/c maybe 3 times since these things were replaced about 6 months ago, so I highly doubt that is the issue. But the weird thing is that the belt slips longer after start up when its hot outside, not cold. I have no idea what could be causing this unless I was sold a bad belt or tensioner.
Does it have an automatic tensioner or you should adjust it manually?
automatic
Change the belt (use ACDelco), if it did again, then go for the tensioner as well.
Probably the AC compressor is bogging it down. The compressor turns on in other modes as well. Not just A/C mode, and no matter the temperature.
You probably should put a new belt on just because this slipping is wearing it out prematurely. 2 things you or your mechanic should check for - oil/coolant leakage on the belt and misalignment of the pulleys.