Maybe the sun gets some kind of feed-back from the temperature of the earth and alters its nuclear reactions accordingly.
In all my years as a Navy Nuclear trained officer, as a civilian with a senior reactor operator's license, in all my reading on fusion and physics, a temperature coefficient of the Earth affecting the sun's fusion rate never crossed my mind nor anybody else's that I've come in contact with.
I think I can safely conclude that is NOT a possible feedback mechanism. I hope after a little more thought you and everyone else will agree too :-)