When i was shooting a wedding a few weeks ago I ran out of battery power, having only had the 1 battery. I've now brought an official spare one so hope to be good to go for at least a days worth of casual shooting. During the wedding there were times where i left the camera switched on, the monitor was folded away and the EVF was off as it wasn't to my eye. I just wondered at the time, and more so now whether in this sort of configuration the GH2 may go to a lower power mode to save battery life, ie. is the sensor perhaps powered down etc.
Hard to know, to my knowledge the manual doesn't mention anything of the sort. It'd be a nice thing to have though, one step closer to that walk around all day with a dSLR switched on and no significant battery drain. But then again with their pentaprisms SLRs have it easy in this regard.
The EVF stays on even when not to the eye. The camera has a power saving mode that's activated when no shots are being taken after a period of minutes I believe. Nothing to do with the LCDs not being used.