Geothermal cooling has been an attractive option in the U.S. Northeast since it emits very little carbon into the atmosphere compared with compressed refrigerant systems which use lots of electricity. Geothermal cooling is based on the fact that the ground basically stays around 50-55 degrees farenheit all year round unless the ground is near a hot spring or volcanic fault line. I would think that the temperature of the earth will warm a little bit which would affect the temperature of the cooling fluid.
I presume that by geothermal cooling you mean geothermal energy. Yes, it's an attractive option and is used a lot in Iceland, but unfortunately there are not many places that have such easy access to it as Iceland. Geothermal energy will be just one source of non-polluting energy, along with hydro-electricity and many others. Today's global warming is just a passing phase, and will be followed sooner or later by another ice age.
The Earth is gradually cooling as geothermal heat is lost into space and the crust is getting thicker, as has already happened on Mars, but this will have no noticeable effect on Earth for many millions of years, by which time we will have gone to join the dinosaurs.