If you have a quality pool table e.g. made entirely from quality hardwoods (no particle-board type woods), and with a slate bed, you will have no issues with extreme cold or extreme warm temperatures.
Your slate may "sweat" if you heat a cold room very quickly, and obviously these temperatures and humidity levels will affect how the cloth will play, but it won't damage the pool table itself.
One caution is that humidity and temperature extremes are bad for wood products. Also you'll want to keep it out of direct sunlight and away from the heat and air-conditioner registers.
Now, if we're talking what is the best optimal temperature for a pool table?" Then we can look to the heated-slate carom tables (assuming they are engineered for the best possible play). From the wikipedia article:
[the bed] is often heated to about 5 degrees C (9 degrees F) above room temperature, which helps to keep moisture out of the cloth to aid the balls rolling and rebounding in a consistent manner, and generally makes a table play faster.
This is all done with the goal of controlling moisture levels in the cloth.
In my opinion, the best thing you can do is keep the moisture levels as low as possible. If you are going to install any climate control equipment at all, make it either an air-conditioner or dehumidifier for humid times.
One other note I came across that is relevant (edited down for length, and emphasized the part which more directly answers your "baseline" question):
I have a pool table in my detached, cinder block structure garage with no "real" insulation, no heat, no cooling system, so it can get really hot during the summer and really cold during the winter (Colorado).
The one thing that I've learned is that the constant extreme changes in temperature does funny things to the rails and the wood. So I learned to try and keep the temperature in the garage as constant as possible. My advice would be to run the AC all of the time, not just when someone is in there.
The first winter and summer were rough on the rails and cloth. After that I've been able to keep it to about a +/- 30 degree fluctuation and it seems to be doing fine.
In this climate I can use a swamp cooler and a small electric heater. I sealed the garage as best I could, instead of real insulation I put area rugs on the floor, hung moving blankets on the walls and over the windows, and in the frame-work above the table I put a bunch of 2X4's and plywood and use the extra space (make-shift attic) as a storage area for boxes of old cloths, books, etc., all of which is pretty good insulation.
Disclaimer: This information is not a professional recommendation or even professional advice. It is meant to help you further your own research, and isn't a guarantee of any kind.
Good luck though, and let us know what you end up doing.