For the most part, any oil will work well in typical outdoor power equipment. Most engines don't accumulate many hours, and don't run all that hard.
However, "IF" your engine is asked to run at full load (full throttle and RPM) in a hot climate, it's always a good idea to consider the viscosity/oil temperature relationship.
I found that both my mower and generator oil temps exceeded 265 deg F during heavy use. A non pressure lubricated engine (splash lube aluminum connecting rod) is at risk with poor performing low viscosity oils. An example of
asking for trouble" would be stationary portable generators, running a house and heating water (4500W heating element) during the summer, on 5W-30 dino oil.