"this stuff absorbs heat...it probably tastes cool and refreshing?" (eying forbidden thermal paste tube)
For those who just need a tl;dr
You cam use toothpaste temporarily. The reason why you shouldnt is a few reasons:
It can be conductive, its not the best conductor of heat, and most importantly, it dries out extremely fast (youd basically be repasting very often)
In order to conduct heat through those gaps since air is a terrible heat conductor
I used mayonez for like 4 days waiting for thermal grease to arrive
Am I really supposed to replace thermal paste every couple years? Can't I just have a thing that continues to work?
Nah, the modern one's work easily 6+ years, change it if your thermals start to raise unexpectedly (and even then dust build up or a broken fan is the more likely problem)
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.
There are also conductive thermal pastes. At least there were a few years ago.
It works until the toothpaste gets cooked. Then it's very hard to clean afterwards.