It will work (so too will using no paste for that matter, though I wouldn't recommend it for heavy loads like gaming). It's just a matter of how easy/hard it will be to clean it off after you get some real paste, and if it will damage any components you accidentally get it on.
The reason toothpaste is often mentioned as opposed to peanut butter, Nutella, etc. is because it's not biodegradable so is stable long-term. You don't need to reapply it unless you see it's starting to dry out. Just be aware that it's abrasive, so you want to wipe it off rather than scrub it off. Scrubbing could potentially leave microscopic gouges in the heat sink and CPU spreader. (Turning gears for a few hours with toothpaste in them is a common trick...