Hawking radiation is predicted using the steady state solution for a black hole which has already formed (this is discussed in some textbooks, such as MTW's Gravitation), but I think it is first important to discuss whether such a steady state solution can appear in our universe.
Because time appears to stop at the Schwarzschild radius, an issue is raised as
to whether a singularity can actually form. In 1939 Julius Robert Oppenheimer
(one of those known as the “father of the atomic bomb” for their role in the
Manhattan Project) and one his students, Hartland Snyder, published the first
calculation of gravitational collapse They concluded that, from the point of
view of an exterior observer,
“it is impossible for a singularity to form in a finite time.”
The Schwarzschild radius was interpreted as a boundary at which time stopped.
Black holes were called frozen stars, never actually becoming singular. They calculated
that an observer on the surface of a collapsing star would see a different
“The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day.”
Thus, a collapsing star will create a singularity in finite proper time for an infalling observer, but infinite time for an outside observer. The interior solution, even if real to an infalling observer, exists only in our infinite future, and cannot exist at all in a
universe with a Big Crunch.