Edit: After some thinking, I feel this would make the game worse rather than better. Thank you for reading.
Introduction
Warmth is a big aspect when it comes to survival. It's what emphasizes the reason to have fires and shelters. In addition, not having the proper clothing at the right time, will kill you.
Purpose
Fire
Imagine it's night, you're huddling up with your friends next to a fire to gain some warmth. You talk about you day and then afterwords go back out into the cold to finish your daily routine. Not only will this give fire another purpose, it will bring your community together. Even a traveler might show up to gain some warmth, chat away, and then be back on their tracks.
Wardrobe
Sometimes it feels like your wardrobe may be a bit empty or lacking purpose. That wouldn't be if leather and fur clothing kept you nice and warm in cold environments so you wouldn't have to make a fire every now and then. That way, you'd have reasons to have different clothing in your wardrobe.
Shelter
So it's raining outside, why not stay inside? No one wants to be wet on a rainy day. Shelters could provide warmth, which gives them a purpose other than keeping out creatures and being a decoration.
Actions
What's a better way to get some warmth than being active? If you're freezing and refuse to build yourself a fire, cut some trees down and build yourself a cabin! Soon or later you'll realize that you're not going to need one after all of your hard work.
Lit Structures
After your swim back to the shore, you decide it's a good idea to bake some food because you're starving. Then you realize, it's a great idea to stand around the oven to dry yourself off so the cold doesn't kill you.
Personal Beliefs
Night
Nighttime warmth reduction is slowed by 15%.
Day
Slowly gain 0.5 warmth when it's day.
Consequences
When you are low on warmth, the player will take minor damage overtime. It is not critical, but eventually you will die from it. Since your cold, your muscles tense up, making you move and even do actions much slower. It is possible to gain warmth back when your cold by moving and doing actions, but it would be quicker and better if you got warmth from a heat source.
Summary
Gained: Inside shelter, actions, clothes, fire, lit structures
Reduced: Swimming, night, weather (if added)
Consequences: Take damage, move slower, slower actions
Basically, the idea is that players will have to watch their warmth or else they will have to deal with the consequences, which would eventually lead to death. Swimming, being outside at night, and being in rainy weather, would reduce your warmth. Having clothes made from leather and fur should stabilize your warmth in a cold environment. This would improve these items and would make a wardrobe more useful as well. However, being in other types of armors would still make you cold in those environments. The night and day personal belief system could also be altered, making you gain warmth at day or slow warmth reduction at night. This would also make night and day phases more interesting.