It's water based, which makes it inherently useless as a real thermal transfer agent, because it'll dry out quite quickly. It may also cause corrosion.
There was nothing besides common sense stopping me from doing a quick test with the stuff, though, so I spread some on the 6Cu+'s base. The toothpaste I chose turned out to be slightly runnier than Arctic Silver 3, in case you care; it's pretty easy to spread it very thin. I put on a thicker layer, though; "optimising heat sink toothpaste application" is not something I intend to put on my curriculum vitae.
Firing up my Minty Fresh Thermal Transfer Experiment produced, rather hilariously, an excellent score - 0.47°C/W.
That's right - Toothpaste Superior To Arctic Silver 3! Film At Eleven!
This was only over half an hour, though. When I removed the cooler from the heater, the toothpaste that'd been squished out around the edge of the contact patch was already dry and crusty, and the rest of the paste would follow soon enough. It'd be just minty white dust inside a week, and probably wouldn't work much better than no thermal compound at all. Maybe worse.
But in the short term, and assuming you don't bridge any contacts with it (it's sufficiently conductive to cause problems), it would seem that toothpaste works just fine as thermal compound. If you're almost out of genuine thermal goop, can't get any more right away, and know you're going to be reseating a processor cooler umpteen times as you fool around with whatever it is with which you are fooling around - well, you can probably make do with toothpaste.