Is it possible to colonize the center of a planet with a cold core?

No.

Reason 1: Even here on Earth there is a problem with mines once they get down to about 7000 feet or so: Creep. The rock slowly deforms under the pressure. Maintaining a tunnel requires ongoing maintenance reaming out the slow flow of rock that tries to squeeze the opening shut.

Reason 2: Useful energy requires a source and a sink. The source has to be at a higher temperature than the sink. Example: We can burn coal to create steam drive a turbine to make electricity. As a heat engine this is high temperature coal fire to low temperature wet steam with some of the difference extracted.

The coal in turn came from the difference between high temperature sun, and low temperature space.

So where does your colony get it's energy.

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That said there are MANY stories (See Poul Anderson's collection, "Tales of the Flying Mountains" about asteroid mining. Energy is external to the asteroids, often solar. Colonizing the surface, or just under of an asteroid makes all kinds of economic sense -- ready source of metals, of oxides, of mass in general. Protection from radiation. But the core of a planet? You need a compelling reason to put up with the immense difficulties.

Comment about lack of structural integrity of asteroids:

Mass+ energy = habitats.

Take a ton of plastic that has a plasticiser that will boil off. Stick a hose in it, and inject ton of gas. Doesn't matter much which. Inflate the plastic until it's a bubble 10 microns thick.

Glue a door to it.

Enter the bubble, move to the centre and boil a pound of aluminum on one side of a large plate. Half the aluminum freezes to the plate, the other half coats half the interior of the bubble. Vapour deposition may not work. There is still some very thin gas in the bubble. Electrostatic?

Anyway now you have a several hundred foot diameter bubble that is silvered on one half. Makes a badly focused mirror. Wanna bet that the focus isn't hot enough to melt most rocks?

Make other bubbles. Put medium sized asteroids in them. Put enough mirrors shining on the rock to boil out the gasses. Collect and store for use later. I'd expect the gas composition to be similar to cometary gasses.

Spin a larger asteroid slowly. Fuse the outer 10 meters of crust into something more solid and durable. Now hollow out the inside.

Why would one want to reach the core of a planet?

To reach it one has to dig a few thousands kilometers of rock, and there would be only dense elements, iron and nickel in the case of Earth, which could be found also more superficial layers. And the more the hole is dug deep, the more gravity is pushing to close it.

I remember reading in the 90s about the Kola deep drill, where they said that samples of rocks taken at the depth of "just" 12 km would explode on their own when taken, just for the sudden release of the pressure.

Here you even have a cold core, so you are not even chasing for the last source of energy in a dying world.

All in all I think the feat is not worth the effort.