Planets loose heat energy over time by emitting radiation.
Correct. And planets also gain heat energy from radioactive decay
(mostly by potassium, uranium, thorium;
see Earth's internal heat budget - Radiogenic heat)
and by absorbing light from the sun.
But is there a limit to how much heat energy a body can loose ? The moon is still hot at the core.
The planet's core and mantle (several 1000 km of iron and rock)
is quite a good heat isolation. Therefore cooling takes a long time
(several billion years).
See also Earth's internal heat budget - Primordial heat.
Neglecting the above mentioned radioactive decay,
cooling of the core will continue until the temperature of the core is the same as the
temperature of the surface (currently somewhere between 0 °C and 40 °C).
I find it hard to imagine a core still under tremendous pressure yet frozen.
There is no contradiction between high pressure and low temperature.
So a cold core will be possible.