No it won't hurt, especially in warm weather.
Why would you want to put in an oil other than recommended? 5w30 and 10w30 have different cold flow characteristics, other than that they function the same and have the EXACT SAME viscosity at operating temperature. Don't try to re-engineer the wheel, use what is recommended as listed in the owner's manual.
You are right
Looking for a Used Cobalt in your area?
CarGurus has 213 nationwide Cobalt listings starting at $1,950.
Search Chevrolet Cobalt Questions
Chevrolet Cobalt Experts
Related Models For Sale