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The gravity of the earth will remain the same, all other things being equal, whether or not the core was hot or cold. Gravity, as Einstein first showed, is a curvature in space-time resulting from mass. Only mass, not temperature, is relevant here.

But what would happen if the liquid metal core of the earth cooled? The magnetic field of the earth would diminish (it is a "dynamo" now, due to the movement of the liquid in the core), as well as stabilize (that is, the north and south poles would stop moving about as they have done for millions of years). We might have some difficulty using common compasses, in this event. Of course, our GPS and celestial navigation would still work fine.

EDIT: The loss of the earths magnetic field would make havoc on earth.

No more magnetism means that the suns solar waves would make atmosphere burn off,( no more AIR!!). The air pressure would diminish from the loss atmosphere. The less air pressure means the boiling point of ALL matter on earth would go down, making it easier for stuff to melt and burn. The oceans would burn off. Earth would be like mars with out an atmosphere. It will happen one day, the earth is heated by radioactive decay in the core. The uranium would eventually be gone. That is why the earth would cool.

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