The amp doesn't run any cooler with a mismatch. That's bogus and they should be ashamed of themselves for saying that.
Go here, to R.G. Keen's tube amp FAQ:
http://www.geofex.com/tubeampfaq/TUBEFAQ.htm#mismatch
The salient points:
It's almost never low impedance that kills an OT, it's too high an impedance.
The power tubes simply refuse to put out all that much more current with a lower-impedance load, so death by overheating with a too-low load is all but impossible - not totally out of the question but extremely unlikely. The power tubes simply get into a loading range where their output power goes down from the mismatched load. At 2:1 lower-than-matched load is not unreasonable at all.
If you do too high a load, the power tubes still limit what they put out, but a second order effect becomes important.
There is magnetic leakage from primary to secondary and between both half-primaries to each other. When the current in the primary is driven to be discontinuous, you get inductive kickback from the leakage inductances in the form of a voltage spike.
This voltage spike can punch through insulation or flash over sockets, and the spike is sitting on top of B+, so it's got a head start for a flashover to ground. If the punchthrough was one time, it wouldn't be a problem, but the burning residues inside the transformer make punchthrough easier at the same point on the next cycle, and eventually erode the insulation to make a conductive path between layers. The sound goes south, and with an intermittent short you can get a permanent short, or the wire can burn though to give you an open there, and now you have a dead transformer.
So how much loading is too high? For a well designed (equals interleaved, tightly coupled, low leakage inductances, like a fine, high quality hifi) OT, you can easily withstand a 2:1 mismatch high.
Note that he says the OT can withstand the spikes...that does NOT mean that the tubes can.
