They do not mind the cold.
Donkeys dislike rain, and are susceptible to pneumonia and bronchitis when chilled.
In Canada during late spring, summer and early fall an open front shed will do for shelter if it is well bedded with dry straw. In winter, depending on the region of Canada, donkeys may be shut in a barn, but allowed to run out on good days, or they may be loosed housed in a comfortable shelter facing away from the prevailing wind. Some donkeys like snow, but others suffer from the cold. Guard against chilling by the wind. Wet snow can melt down into a donkey's coat, soaking the hair and causing the animal to chill.