Holding in lower gear vs letting tow-haul mode shift

wayfun wrote:
Intuitively I feel less shifting would be better but I've been told that allowing it to shift would make the transmission run cooler.

Actually, unless the torque converter immediately locks up, frequent shifting will result in more heat.

If it is hunting for a gear and shifting on every grade and won't stay in any one gear for more than a couple of miles I would lock it in 4th.