I have not seen any scientific evidence of cuticle closing from being washed with cold water. What I think happens is cold water doesn't strip off conditioner or whatever it is you just used on your hair the way warm water might, which is why your hair might feel softer. But if you had bare hair and washed it with cold water and then bare hair and washed it with warm, I don't think you'd have softer hair from the cold water.
I went to boarding school and for years washed my hair with cold water. I never had soft/smooth hair from that.
So until someone shows me some scientific proof that cold water closes cuticles I will not believe it does, just like I do not believe cold water closes pores on the skin. The most cold water does to skin is make blood vessels pull away from the skin.
If cold water was the cuticle closer you all make it out to be, my hair should be the most tangled because I never use it. Today I didn't even ACV rinse because I was in a rush, yet I was just redoing a twist and it was so easy to separate strands--something I believe would only be possible if the strands were smooth. Did I mention I didn't apply anything to my hair after the wash?
ETA: And if temperature played a role in opening and closing of cuticles, then it would be a very essential part in relaxing hair. As we all know it is pH and not temperature that is called to play when hair is relaxed. Need I say more?
This rumor about cold water closing pores seemed to have started the same way as the rumor that face exercises cause wrinkles did. Someone just THOUGHT it made sense and suddenly it became the "gospel". And like a placebo pill, because you buy into it, you believe it without considering other factors like how rinsing a greasy plate with cold water might not leave it as "clean" as rinsing it with hot water.