I think, all things considered, the chances of the Earth's center cooling to the point our planet becoming cold and dead are pretty slim... We know stars (of which our sun is a fairly middle aged example) become much larger as they begin to die which would result in our planet (and nearly all of the solar system) becoming engulfed in the red ball giant expanding to turn us into a cinder.
Additionally, this from the "New Scientist" indicates Mars did and still does have a liquid, molten center:
"It has been known since 2003 that at least part of Mars' interior is molten, based on how easily the Sun's gravity distorts the planet's shape, but no one knew whether it is completely liquid, or whether there is a solid inner core like Earth's.
Now a team of scientists, led by Andrew Stewart of the Swiss Federal Institute of Technology in Zurich, Switzerland, has succeeded in creating in the lab the high pressure and temperature expected in Mars' core..."