Thanks for the response. I finally hooked in a separate transformer. I had to cut the cable from the thermostat to the HVAC. It was a six wire cable, with 5 active for the stat. The red was power and blue was common. I hooked these two to the transformer but the HVAC did not seem to go on (the 2441 stat powered up fine but the heat and cool setting were not powering up the HVAC). I phoned Smarthome, who suggested I run only the power to the transformer and run the common just to the HVAC unit. That produced no power, and somehow, the power from the HVAC terminated. I could not get the 2441TH to turn on connecting the wires as they had been. I checked with a voltmeter and there was no power getting there. Somehow either my power or common got shot as I was working on this. I reconnected the transformer, so the red lead connects to transformer, 2441 and the HVAC red, and the blue lead connects to transformer, 2441 and the HVAC blue. That seems to have the system running.
I will wait a few days and see if energy mode and/or cool cycle kicks in on its own (tends to happen infrequently, not regularly, and when it is colder than it is right now). I have no idea what happened to the power from the HVAC itself. I see no evidence of a wire coming loose.
I am going to try this:
http://www.apc.com/resource/include/techspec_index.cfm?base_sku=pnet1gb&tab=features