Newer engines with full pressure oil systems will work best on multi-weight oils, and engines with a splash system work better with a straight weight.
Kawasaki recommends either a 10-30 or 10-40 weight oil, and I would probably go with the 10-40 being it will be used in extreme heat.
If the engine is new, you can use synthetic oil, but I would break it in with conventional oil first. If it's an older engine with many hours on it, I wouldn't switch to synthetic. At your first oil change would be the time to switch to synthetic. Synthetic oil withstands heat much better. I've seen some tests show them going strong a hundred degrees after mineral based oils had failed.
A 10-40 synthetic should give you the best protection during the summer in your mower.