Exercise after FUT hair transplant, will wearing a hat to class do more damage?

You need to take it easy the first week, as your surgeon likely told you.  If you go right back to work you will make money but will compromise your results and likely void any guarantees you were given.  Better to protect your investment and follow your postop instructions to the letter.

To the general reader: A FUT is a kind of hair transplant where a strip of hair and skin is removed from the back of the head. This donor area is sewn closed. From the strip, follicles are dissected and placed in the recipient site to either lower the hairline or improve density.To you: Healing from FUT involves rest. You must allow the back of the head to remain tension free to avoid having a widened scar. The follicles in the recipient zone are held by scabs, not sutures. You do not want to sweat, raise your blood pressure, or mechanically disrupt the follicle with exercise, rubbing, or a hat.  You can wear a hat so long as it does not contact the follicles.  Additionally, gyms are rife with bacteria. I would urge you to make a small sacrifice in pay this week or do the bookkeeping/front desk/back end part of the job in order to optimize your result.  Best of luck!