There are two possible causes.
Pre-existing underlying conditions that cause wrist pain, such as tenosynovitis, rheumatoid arthritis, fractures, local contusions, etc., may intensify the pain when stimulated by cold.
Prolonged exposure to air conditioning may cause transient muscle spasms, which in turn may cause pain. It is advisable to control the time and temperature at which the room air conditioner is activated, so that the time is not too long and the temperature is not too low. Clothing can be added in moderation to protect the painful area for warmth and oral anti-inflammatory and analgesic medication can be taken to reduce pain if necessary.