The issue with air cooled generators is that they have no way to manage oil temps under extreme conditions. An example of which may be after a tropical storm power outage. When it's really hot, humid, and you place high demands on the generator (A/C units, water heater, well pump, etc)
What FL generator users discovered is that there was little air movement around the units, temps climbed until the conventional oils were unable to protect the connecting rod bearing and metal to metal contact resulted in a broken connecting rod, or wildly worn out rod. (remember, there is no bearing there, just an aluminum rod on the crankshaft)
I had 2 Honda water pump engines fail on 10w30 conventional Pennzoil. They simply overheated the oil, running 24/7, with oil changes every 2 days. Switched to 15W-50 Mobil 1 and problem solved.
Co-worker's generator failed by the very same method. 10w30 oil, hot conditions, running a 4500W water heater element so mama could have hot showers. He's an aircraft mechanic, and did not let the oil level run down.
My suggestion is this: Use a quality, heavy duty oil (I prefer synthetics for the excellent high temp stability) , with a viscosity appropriate for the temperature outside.
Good choices include Mobil 1 5W-40 Turbo Diesel Truck (a very robust oil and my all time favorite due to so many excellent UOA results and real world results)
Rotella T-6 5W-40, Mobil 1 15W-50 or the 10W-40, Mobil 1 Motorcycle 20w50 oils. Just about any Diesel engine oil, and of course, Amsoil's synthetic small engine oil.