I don't really understand the whole sun dying out hypothetical, I mean if the sun died out everyone would die pretty fast because all of our crops would die and we wouldn't have any food, especially in counties that rely on rice and stuff like that for main crops. And even in the more civilized countries solar energy is a big source of energy, so that would help a collapse of civilization. Also darker people don't need the sun (as much as you claim) to survive, they aren't flowers, the higher melanin content merely protects the skin from UV rays, giving them a lesser chance of skin cancer. The only thing that would happen is we would get less Vitamin D and we would have to get it from another source, assuming our crops aren't dead and we haven't been incinerated from the explosion of the sun...?
And assuming we somehow all survive from there being no sun, then our skin would just adapt through microevolution, over years and generations to be paler and more susceptible to some sort of sunlight from some other place, we wouldn't die. D: Besides last time I checked skin color didn't have THAT much effect on heat, and with all the dead crops can't people just make leaf clothing? (And isn't the theory that darker colors -attract- heat and lighter colors deflect it?)(Melanin, not melatonin reacts to light/radiation, not heat)
PS - If the sun moved close enough to the earth that all the white people ended up dying of skin cancer, then I'm sure it would end up with all of the darker colored people dying too because darker skin only means more resistance, not exactly much better protection. lol and if a disease manages to kill most of the world, I would bet it is airborne and doesn't care about your race if it's going to kill you, being as all humans are 99.99 genetically the same anyway. (Unless it is somehow a melanin targeting virus, which I find to be unlikely and would kill everyone anyway because everyone has melanin)