The coldest brown dwarfs are around room temperature, and therefore pretty darn close to black in visible light - but they nevertheless support deuterium fusion. Remember the temperature of the radiative surface doesn't have to be the same as the temperature in the region where fusion takes place. (That also applies to the temperature of atmospheres discussion up-thread - there is an important difference between heat and temperature and a very very thin atmosphere can have a very high temperature yet still carry almost no heat.)
I'm no physicist, but it is my understanding that after a certain size a star has to have fusion happening inside or it will collapse under its own gravity. I don't remember what the maximum size is, but I would guess that somewhere between Jupiter and maybe about twice as massive, give or take. And, of course, once the fusion reaction is kickstarted, its strength is pretty directly correlated to the mass of the star, so the more massive the star, the hotter (which is why cold stars tend to be very small, much smaller than our Sun, and very hot stars tend to be gigantic.)
In this case the "black" star in the screenshot is listed to have 116 solar masses, so I doubt it's physically possible for it to be that cold (its surface temperature is listed as 31 kelvin). There may be relatively cold stars out there, but I would guess they are very small, much smaller than the Sun.
I realise these things like in your screenshot are more likely a Stellar Forge issue though - where is this one?
The screenshot is from the Phimbeau AA-A h76 system.
There's another famous black star in the Splojeia AA-A h19 system (in this case with a surface temperature of 113 kelvin), if you want to make a tour.
