The Battery Sergeant
I find something of the same with my main portable flash - the Nikon SB 700. It fortunately uses AA batteries and I can put lithium cells in it, but even here there is a real problem of the cells getting hot as they discharge and somewhat slowing down. In the middle of a dance show I have to eject one set and let it cool on the floor while a fresh cold set go it. Granted you can get 600+ flashes from a set of batteries but you have to juggle things.
I was struck by a thought about the design of the guns and the type of battery. Would it not have been better for the manufacturer of the flash gun to make it take the same sort of rechargeable lithium cell that goes into the cameras? I am thinking of a flat pack like the EN-EL series of batteries. Then a photographer could keep a large set of charged cells for both camera and flash and quickly slip them in or out of the flash body.
This would have the advantage of selling more rechargeable lithium cells - and not losing this business to the general battery manufacturer. And of adding the sale of a charger to each flash gun.
Admittedly, that would have provided the general public with yet another opportunity to leave a charger in your hotel room when you check out. And another aftermarket sale for us...