If you run your equipment at 80+ degrees room temp, you will severly shorten the life of any rack mounted equipment even faster than towers because the dense packing reduces air flow, plus blowing 80+ degree air across hot processors is really not very efficient!

If the room temprature is 80+, then the equipment is much hotter than that, especially inside the case. Keep the room Temp at 65-70 degrees, max. That will keep you equipment happy.
First problem you will probably see will be failures in your UPS batteries, as this will shorten their life considerably. Next you will start having drive failures from overheated bearings, then probably other electronic parts like your very "cheap" CPU units. CPUs last a litte longer than the drives because they have their own fans, but they also die from overheating, unless your systems shut down from thrermal overload.
The other failure will be the temper flairups from the IT people having to work in this condition.

Besides, it is not good to drip sweat onto electronic equipment!
As for convincing your boss, here is the path to take:
1. Remind him what this equipment costs to replace, and the time it will take to repair or rebuild each system if you suffer multiple drive failures, etc. (Good for the justification for backup tapes also!)

2. Ask him how long they can operate when the system goes down, and what it will cost them when no one can get mail or do any work for an extended time (READ: DAYS)while you try to order, receive and rebuild from scratch.
3. A thermal failure usually results in one or more failed components and many degraded parts that usually work initially, then promptly fail also once the other failed parts are replaced and a full electrical load is restored to degraded power supplies, etc. Result: Extended, and repeated, downtime!
4. Compare the AC cost to the lost manhours when no one can work.
GET a 24x7 AC unit dedicated only to the computer room, as many buildings shut off the building AC in the evenings and on weekends. Just because you are not there to check, it does not mean it has not gotten very hot in the room!
HTH,
David