The whole "5w-20" and oil consumption is a function of the brand and base stock, not the viscosity itself.
An inexpensive lube that is a lower end grp II+ may have an issue. Whereas a very high quality group IV or III with superior chemistry (such as HPL) won't evaporate nearly as bad as some of the lower tier choices. It has nothing to do with the vis rating; it has everything to do with the base stocks and additives.
If you want something readily available at the local WM or autoparts place, then perhaps consider Mobil 1 in your choice of vis.
If you want something more boutique, I'd suggest Amsoil or HPL; easily ordered, but will cost more.
Generators are a piece of emergency equipment in the manner the OP speaks of. IMO they are worth spending more money on the lube, because the failure of that piece of equipment won't end well for anyone. When the power is out, and you're in the dark, you NEED that unit to run reliably.
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To follow up on my experience I described from a while back ...
As I mentioned I had two generators; one smaller 4200W for the full-time power (furnace; fridge; sump pump;etc), and one 5000W loud ol'skool one for running the well pump. When that uber-cold 3+day cold snap happened, I learned a valuable lesson:
- The smaller genny stayed in the heated garage all the time. It was easy to move out and then pull start. Turned over fine and fired up quickly, because the ambient in the garage was probably 40F due to the small propane wall heater. 40F was enough to keep the genny "ready" for use. This is the genny that ran non-stop for 3.5 days. Once I got it running, it stayed running for about 90 hours. It had 5w-30 syn Mobil 1 in it.
- The larger unit, OTOH, was out in the barn (-17F). It had synthetic 10w-30 oil in it, but it was grp III (Supertech as I recall). When I pulled the start cord, it was so stiff from the cold oil that I could not pull the cord fast enough to get it to fire; the parasitic drag of cold oil was like frozen molasses in that crankcase. I am no weakling, and that cord was HARD to pull to turn the crank over. So what I did was move it into the heated garage and after a few hours, the Tecumseh engine had warmed up enough to warm the oil, and then I was able to get it to fire off. Once I was done with it, it went back into the garage.
Oddly enough, my neighbor had his genny filled with conventional 10w-40; it wouldn't turn over at all, as if the crank was just glued into the case. It sat overnight in my heated garage, and was able to start the next morning. Once started, we got it warm enough to drain the oil, and then put in some 5w-30 PAO.
My point is that when you NEED power, and you're relying on a small, air-cooled genny, you want a quality synthetic oil in the crankcase, because if you have bad luck like me, you'll need that genny at the worst possible time. It's either stupid cold or ridiculously hot, and the last thing you want to worry about is having to do an OCI to get it running. If you think for one second that a really cold start is in the future of your air-cooled genny, then I HIGHLY recommend using a 5w-30 or 0w-30 of a group IV or V base-stock, so that the oil will be thin enough that you can actually get the crankshaft to turn over fast enough to get good spark and also a good draw through the carb.