Earlier quoted context omitted.
The difference is that for solid-state logic, the heat is purely a waste byproduct, whereas for thermionic valves, the heat is the energy used by the electrons to move between the electrodes, hence the name. If the valves are cold, you have to wait for them to warm up before they can do anything.
All the same, "cold boot" doesn't refer to letting tubes cool down. If power is restored while the tubes are still hot, it's still a "cold boot". The electrons are not moved by heat; they are moved by the electric potential between the electrodes. The heat goes into maintaining the cathode at a certain temperature , at which the electrons "boil off" in sufficient numbers, but that requires only a smidgeon of the ener…
There's a lot of physics and engineering behind thermionic emission in a typical vacuum tube and the subject shouldn't be overlooked just because tubes are less commonplace than they once were.
Until the transistor became popular in the latter half of the 20th C. essentially all of electronics relied upon thermionic emission so much effort went into understanding it. Moreover, it's still very important in specialized areas such as in scientific instruments, high power transmitting tubes, TWTs (Traveling Wave Tubes), CRTs, etc.
Thermionic emission obeys Richardson's Law—after physicist Owen Richardson who received the Nobel Prize for his work in 1928, and its study is an essential part of thermodynamics: https://en.m.wikipedia.org/wiki/Thermionic_emission.
When I was learning about this stuff I had a little book titled Thermionic vacuum tubes and their applications by Sir Edward Appleton (who is more famous for his wartime research into radar). Still, for anyone who's interested in thermionic emission then this could be a useful reference.
I'd add that when it comes to practical methods of manufacturing thermionic cathodes, info on low work function oxide emitters etc. then there are references that are better or more practically orientated (but offhand I cannot remember the names of any).