If all stars suddenly stopped producing heat (or heating up) which stars would cool to absolute zero first and how long would it take?
In our current universe white dwarfs are the first ones that should cool, because they are already "cold" (not producing anything anymore, just radiating heat) remnants of a former star. The time for this to happen is disputed (10^15 or 10^37 years), but is far bigger than the age of universe, so nobody expects to find one "cooled star" yet. See this article for details on "black dwarfs" as they are called.
In your sudden stop hypothetical universe probably the first one to cool would be the smallest and coolest stars, a M9V red dwarf (with 7.5% solar mass, 8% solar radius, 2,300K temperature). Note that brown dwarfs are categorized as substellar objects, so they shouldn't be considered. I don't think the poorly understood "Mpemba effect" can be yet applied to stars.