I think it'll be fine. I did a little experiment last winter. I use a programmable thermostat at home, which drops temps to 55 when I'm not at home and then up to 65 in the evening and back down again at night.
I brewed a double IPA. When it was bottled, I brought a sixer to the office where it's always 70ish, and left it in a cabinet for a month. Then I tried a home-carbed beer side by side with the office carbed beer and they were identical. No problems with carbing in the chilly house.
Of course, YMMV.
Who is this Rorschach guy? And why did he paint so many pictures of my parents fighting?