I have never heard of that before.
Hello,
Birth control pills do not make the vagina cold. If there is any such symptom then alternate birth control options like birth control ring can be used.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.