Taking moderate cold water is okay. Excessive cold water from fridge and icy water may give you sore throat, cold and cough. The cold water has same component as normal. So, it will not add to fat.
I haven't heard that too cold water adds ft to body... but it has bad effects on body...
1. It irritates the throat and causes cold and cough.
2. Extreme cold or hot items would damage the enamel on the teeth and damages them.
Ayurveda has advised against the excessive use of cold water, it is retained in the body by cells and may result in swelling hence it is advisable to use only lukewarm water for drinking. in hot weather use of water cooled in earthern pots is not harmful!
Drinking cold water is certainly not good . Cold water often attracts fever and cold . But about getting fat from cold water I don't think it is true .