I say, and with God's success: The fear of cold is not a valid excuse for performing tayammum instead of wudu; because this excuse is specific to ghusl only. It was narrated by Amr ibn al-Aas : "I had a wet dream on a cold night during the battle of 'Dhat al-Salasil', and I feared that if I took a bath, I would perish, so I performed tayammum and then prayed Fajr with my companions. They mentioned this to the Prophet , and he said: O Amr, did you pray with your companions while you were in a state of janabah? I informed him of what prevented me from bathing, hoping that I heard that Allah says: {And do not kill yourselves} (An-Nisa: 29), and the Messenger of Allah  laughed." This is found in Al-Mustadrak 1: 285, Sunan al-Sughra 1: 185, Sunan al-Bayhaqi al-Kabir 1: 225, Sunan al-Daraqutni 1: 178, Sunan Abu Dawood 1: 92, and Musnad Ahmad 4: 203. See: Fath Bab al-Ianayah 1: 110, and Allah knows best.