Big John, you are using this for mowing in the summer as well?
0W30 or 5W30 synthetic in the winter, 10W30 (conventional or synthetic) in the summer. Both "High Mileage" formulas if you can find 'em ... again, to benefit the older seals. If, after a season or two, you don't have any leaks, switch to a HDEO for the added detergency.
I see no reason whatsoever to drop to a 20 weight oil, regardless of the season.
Be sure to grease that thing thoroughly, especially the spindles and the mower deck. It might take you some time to find all the zerks. Some of those critters like to hide. It looks like you have a manual ... do they pinpoint the zerk fitting locations?