why does air conditioning work for a long while then stops and after turning motor off for awhile it starts working again?

Evaporator may be icing up and prevent air flow. I have seen one completely covered with ice on a humid day. It thaws and works again

Sorry, follow up question, not answer to an old issue... OP, did you find an answer? I have the same issue. It is not icing up. I know some people won't believe me, but I have tried this over and over to prove the relationship and use this to minimize the issue. If the alternator disengages and the voltage falls to just battery level, then the A/C cuts out. If I leave the headlights on full time, then the alternator stays engaged, and the A/C never cuts out. Something with the voltage or alternator engagement... something is triggering the compressor to cut out. The A/C behaves as if nothing is wrong, blower does not change, etc. The air just starts to get warm, and then when the alternator kicks in later it starts to cool back off. FYI, turning the vehicle off and then back on resolves it temporarily because the alternator always is engaged for a period of time after each start.