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.
we would die
If the earth got pulled in closer to the sun, or if the sun expands significantly, then the earth will get much warmer. That said, nothing like that is going to happen for a couple billion years yet.
we would probably die
It would get bigger
The sun would be cooler
We would freeze to death
Everything not bolted down, including the atmosphere, would fly off into space, the earth would depart from the sun. The sun and all the stars would explode.
Currently the gravitation pull of the Sun is balanced by the momentum the planets have due to their velocities. If the Planets were to slow down they would be pulled closer to the Sun. They could be pulled into the Sun or regain stable orbits as the Sun's gravity increased their speeds again.
If the surface of the sun cooled, its appearance would likely dim and become less bright. This could result in a decrease in the sun's overall luminosity and potentially affect its color, possibly shifting towards a redder hue.
Nothing realy really. If it sped up it would spin faster and make shorter days and would revolve around the sun faster. If it slowed down it would spin slower making longer days and it would revolve around the sun slower.
The sun would win.
It would melt.
You would be vaporised long before you actually reached the sun.
death
It cannot happen, but if it did we would all die shortly therafter.
you will die
we would die