Cold shouldn't hurt anything, and given that it's going to be inside, I doubt that the inside would get hot enough to be a problem. I think that wood stocks would be the most at-risk item in the safe. Humidity is probably a bigger issue than heat.
There are a variety of ways to control humidity within a safe that range from disposible bags of whatever-that-moisture-absorbent-powder is called, to electric powered dehumidifiers that you will need to run a cord into the safe to use.
I got a small canister of that powder that I "recharge" in the oven when it needs it (as indicated by a small blue dot in it). Which option is best for you would depend on how big a safe you have, and what you're going to store in it.
Even in the worse case, I think if you perform "routine" cleaning and oiling of whatever is in the safe you'd prevent any serious harm. Depending on how extreme the conditions are in the safe, and how valuable the firearms are to you, would determine how often that that routine maintenace would have to be doen.