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Stars radiate heat, not cold. e=mc^2 doesn't say anything about cold... But when the energy all leaves at speed of light, what's left over is very heavy cold mass and it will sink deeper toward hole formation.
Neutron stars are formed by some of the same processes as create black holes - just in less massive stars.Young Neutron stars have a temperature around 1 million degrees (and billions at first formation). They are not formed by a blob of cold sinking to the center of the star!See: https://en.wikipedia.org/wiki/Neutron_star#Mass_and_temperature
The difference with a black hole is that the gravity is so intense that even the light and heat from the initial fireball can't escape from the black hole.
The exterior of quiescent black holes do have a temperature, due to Hawking radiation, but for stellar-mass black holes it is incredibly close to absolute zero (nanoKelvins). See: https://en.wikipedia.org/wiki/Hawking_radiation#Overview