Stirling AC, someone tell me why it wouldnt work. Answered
Take a split type airconitioner unit (with heat exchanger / motor etc outside the house) remember that heat pumps have COPs >1 so they remove more energy than you put in. Now take a reasonably efficient stirling engine and bolt it onto the external heat exchanger so there is a gradient between the hot heat exchanger and the warm air outside and use it to pump the refrigerant around the system. Surely a heat-pump stirling engine combo can be made efficiently enough to effectively run itself once it has been started up, maybe with a few watts of electric to run some fans.