I leave mine plugged in all the time - using the extender and AC power. Sure it gets warm - so does most of my electrical equipment, even when running on standby.
It should be pointed out that leaving it plugged in does not guarantee to make it any more likely that my Chromecast app will find the device the next time I need it to, but that's an issue for an already long thread elsewhere...
Meanwhile, there's the incessant images that are being downloaded, one assumes, even when I've not got the TV on. They're unnecessary, 'something shiny' and probably drawing on my internet cap, however marginally - but hey, its not costing Google anything, is it?