The winter season is around the corner and I am wondering if it's damaging to leave tefillin out in the cold (I usually leave mine in a room with no heat).
Would this ruin the tefillin because of possible shrinkage?
Extremes in temperature, both heat and cold, commonly cause several different types of damage to tefillin:
The batim can crack. The paint of the batim and the retzuot, straps, can shatter. Even the ink of the writing can crack or split away from the parchment inside the batim when you're not wearing them, store your tefillin at room temperature and away from direct sunlight.
I once left my tefillin in my car in the middle of the Summer. I frantically took them to a sofer in Baltimore to get checked. He was very amused at my being worried. He told me that it is much worse to leave them in the cold than in the heat, but that neither are much to worry about with the quality of today's tefillin.
(I had him check them anyways, and it turned out they were pasul for several unrelated reasons, and I had to get new tefillin.)