If you set the tub in mortar, you essentially provide a very solid foundation for the tub, and (at least the bottom) feels much more solid. It will usually last a little longer before it gets stress cracks as well.
I've not done it, but my guess is that you could probably spray some foam sealant on the tub to help to hold in the heat. I'm not a pro, and haven't tried it. It shouldn't hurt things as long as you leave room to get to the drain and overflow assembly. that would add some insulation and perceived solidity to the tub as well.