Have early Colt 1/12 thin profile barrel, free float tube, and flat top.

Would like to make into a varmit rig.

Will the thin barrel heat up if I put 5 or 6 quick rounds through it and the point of impact start to wander?

No, you would have to put at least several hundred rounds, a la "full auto" through it before you would see or notice barrel warp.
Thinner barrel heats, but also dissipates heat faster.
Remember, these weapons were made to take it, So shoot it like you stole it!
Edited to add:
If you really want good accuracy you may want to rebarrel your rifle to a tighter twist rate, Maybe a 1 in 9, helps on stablizing the bullet.
Also gives you a wider choice of bullet weight.

Actualy I was thinking of building and upper just like this only I would probably go with a little faster twist as GS808 suggests.  Just use lighter bullets at higher velocities.

The POI may/will wander a little as a smaller barrel won't dissipate the heat as rapidly as a heavey barrel.  Less surface area to move heat into the air around the barrel.  I was going to use one of those aluminum one piece free float hand guards that is also the barrel nut and drill or mill ventilating slots and or holes all along it.  

6 fast rounds and leave the bolt back and it should cool down fast.