All of the oils that people recommended on this thread are good oils for your genset. GC will work year round, so would T6, and M1 0W-40 TDT.
Your genset probably has a heater that stays on in the winter to keep the oil warm, and if so the Amsoil Z-Rod 10W-30 ZRT would also be a decent choice, but why settle for a 10W for winter use when you can have a 5 or a 0.
GC will hold up to summer heat, and it is great in the winter, I have M1 Vtwin 20-50 for 100 F days, and it is probably a better choice for those kind of days, but GC would work.
One thing I would NEVER do is use any 20W-50 below 15 F, and below 75 F you are probably better off with something thinner.