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Can neutron stars die?
Yes, if you throw them hard enough
If cooling down counts as dying. They don't produce heat or have much activity at all
They're still neutron stars though. As far as we know the only ways for a neutron star to stop being a neutron star is for enough stuff to fall into them that they collapse into a black hole, two merge to become a black hole, or if they get eaten by a black hole. Otherwise neutron stars are just about the most stable things in the universe.
Interestingly enough, the inner Layers of a neutron star are commonly modeled to have a temperature of 0K, since the extreme density prevents thermal movements
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