Can I build up immunity to cold temperature?

Immunity is a strong word.
You can acclimate to different environments and temperatures. Brown fat tissue can be increased when exposed to cold temperatures over a long period of time, in a form of adaptation. This type of fat helps produce heat, keeping you warmer.
https://en.wikipedia.org/wiki/Brown_adipose_tissue
There is truth to the jokes of Minnesotans wearing shorts on the same day Floridian's would wear winter clothes.
- Yeah I wouldn't use the word immunity, I'd say you can build up a tolerance.
- Just backing up all of this. I'm from PA and I remember one year we spent Christmas in Tampa. First time I could wear shorts outside on Christmas, while everyone else was sporting pants and at least a hoodie, if not a coat.
  Just thought I should add that this was in the 90s, climate change my effect your results.
  
  I grew up in a northern state and used to pride myself as a teen that I could wear a t shirt in the snow or be wearing a hoodie at the busstop while others had winter coats.
  After a some years in Southern Florida a slight breeze bringing it to 70 degrees gets me into a sweater lol
  The body really acclimates. It used to be so hot the first summer and nowadays it's not a big deal. But actually this summer has been unusually hot, however that seems a bit anomalous.
  
  I'm also from PA and moved to a southern state. It's fun when it's 60 degrees out and everyone is walking around in winter jackets and scarves and I'm in a t-shirt literally dripping sweat. My first winter it was 55 out and someone said "I can't wait for spring to come, I'm so sick of the cold." I just looked at them and said "uh... where I come from, this is spring."
- A couple of years ago for Christmas I left my 80°F December to visit my in laws where the highs were 30°F. I had absolutely no resistance, and it was miserable. They kept their house cold because they were acclimated. I was shivering in my jacket next to the fire the whole trip.
- I am that Minnesota! I've been shorts and t-shirt during winter since high school.

An immunity? Fuck no.
A tolerance though? Yes, you totally can!
You'll never be invulnerable to cold of course, but you will adapt to it if you're exposed regularly to cold.
The real trick to dealing with extreme cold is layering up with the right gear for the conditions, but being adapted to it definitely helps!

You can build up resistance. I live in WA, when I visited Spain everyone was shocked to see me perfectly fine in just a t-shirt when it was 55 american degrees out
- I've never seen someone refer to Fahrenheit as American degrees, but it makes sense
- If you were only wearing a t-shirt, I'm betting people were more shocked at your lack of pants than your tolerance for the temperature.
- 55 american degrees
  That's 13 normal degrees, for the rest of us.

Be careful about the limits of adaptation. You can certainly adjust to temperatures that are colder than you're used to. You will still die if you are inactive outside without protection below 50 degrees F (10C).
- You have to get at it to die from 10°C IMO. Zero for sure, 3h sleep and you won't wake up.
  
  Funny mental image of man huffing and puffing trying to kill himself in a snowbank thats more mud than snow.

Yes, you can build tolerance to cold temperatures by slow gradual exposure to them. Using cold water at the end of showering and wearing one Layer of clothes less to feel slightly cool (but not freezing) works well. Aside of cold tolerance, the benefits include boosting your imune system, turning your white fat into Brown fat, which is able to burn calories (and make heat from them, that's why you feel warmer) and lowering inflammation in your body.

You can definitely build tolerance. So long as it’s above 50°F, I’m likely to be shirtless while hiking or running. Wim Hof may be onto something in terms of what’s possible.

I moved from a hot part of the world to a colder part. I'd say I've climatized and acclimated to the colder environment.

Trying to become an Aes Sedai?

You guys have Spring?!? We just go from Winter in May to Summer in June!
- From frozen hellscape to road construction

Run 5ks at dawn in the winter and you'll never feel cold again.
- I’d feel I was cutting my 21k runs through the snow and ice short though.
  And I’m fully covered on those; no shorts and Tshirt until March-ish.
- That never worked for me. I used to run in the early morning before dawn and if it was warm I could warm up and feel good. If it was cold (yes sometimes it gets cold even in Florida) I had to run wearing layers to be able to warm up enough to feel safe running. Sweatpants over yoga pants, tank, long sleeve running shirt, jacket.
  I have seen people adapt to the heat here though, and my brothers who grew up in warm climate and moved to the northeast certainly adapted.

Look into wim hof. He's probably the best in the world.

Yes.
https://www.wimhofmethod.com/cold-therapy
- Approach with healthy skepticism. That page and its links to other studies are not enough to support its stated claims or implied benefits. https://sciencebasedmedicine.org/wim-hof-the-iceman/ does a better job than I can when it comes to the details. Looks like snake oil to me.
  
  Oh, I wouldn't ever suggest that the method does everything its author says it does - the claims of helping in cancer cases and such are wee bit too "optimistic".
  But the question was concerning the possibility to build up cold immunity. And the answer is "yes". The method I mentioned absolutely will build up cold's immunity (or resistance, or tolerance if language purism is a factor) in everyone who will attempt it.
  There's no magic hocus-pocus in it. It's very simple concept used in many "methods" - Buteyko comes to mind, so does Russian Siberia.