Hi! My 5 yr old boy has trouble with temp changes as well, not to this extent though, and usually his is hot (if the water is too hot, the sun looks at him the wrong way, his food is too warm, you name it!) My experience with him is that he would say it hurts because that is the language he had at the time; but that part I cannot say for sure, because temperature changes can experienced differently to those with SPD. I have had similar instances with my son that you experienced with yours. I would guess that when your son got out of the water, the change in the air temperature he was unable to process- I went through this with my son. (We have had some interesting moments!)
I have found that knowing that J is sensitive to warmer temperatures as well as temperature changes is helpful in that I can be proactive. I make sure I put his dinner in the freezer before I give it to him; the bath temperature is consistant, we cool his carseat before he gets in, in the summer I keep hats in the car for him, etc. It doesn't always work at preventing meltdowns, but it can be helpful.