Why does the heat energy given off by an object increase as the object
gets closer to ambient temperature?
Ex: Drop a penny into liquid nitrogen and it bubbles as the loss of
heat energy causes a phase change in the liquid nitrogen, but just
before the bubbling stops it increases. Why?

Is this the good old Leidenfrost effect at work, I wonder?
The production of nitrogen bubbles will actually insulate the metal
from the bulk of the liquid until the temperatures are nearly
equilibrated.

This suggestion is an excellent one but it doesn?t quite fit. The
Leidenfrost effect has mostly to do with insulation such as dipping
your hand in cryogenic gas, hundreds of degrees below zero and not
hurting them. The situation I am interested in is the movement of
thermal energy from one material into another. It is easiest to see in
any gas with a boiling point below negative 100 degrees.  When you
drop a metal object into this liquid it bubbles as the heat energy
moves from the metal to the liquid and causes the phase change. It
looks like a Fizzes tablet dissolving in water except the rate of
bubbles is the same until just before they stop. Then it is much like
the storm before the calm. It is this storm or it?s cause that I am
looking for.