This B&S response to engine oil viscosity
Yes this different than Kohler's recommendation.
Small Engine Oil Type Recommendations:
Use SAE 30 in warmer temperatures of 40° F and higher (5° C and higher).
Use SAE 10W-30 for a varying temperature range of 0 to 100° F (-18 to 38° C), this grade of oil improves cold weather starting, but may increase
oil consumption at 80° F(27° C) or higher.
A synthetic oil SAE 5W-30 is the best for very cold temperatures of -20 to as high as 120° F (-30 to 40° C) providing the best protection at all
temperatures as well as improved starting with less oil consumption.
Use SAE 5W-30 for very cold temperatures of 40° F and below (5° C and below).
For continuous-use, such as commercial lawn cutting or pressure washing, use Vanguard 15W-50 Oil (temperatures from 20°F to 130°F).
Things to keep in mind when choosing lawn mower oil:
Use a high quality detergent oil classified "For Service SF, SG, SH, SJ" or higher.
Do not use special additives.
Synthetic oils are an acceptable oil at all temperatures. Use of synthetic oil does not alter required oil change intervals.