I had nothing in frosty surroundings - that's stupidity (6)
I believe the answer is:
idiocy
'stupidity' is the definition.
(I've seen this before)
'i had nothing in frosty surroundings' is the wordplay.
'i had' becomes 'id' (I'd).
'nothing in' becomes 'o' (I've seen this in other clues).
'frosty' becomes 'icy' ('icy' can be a synonym of 'frosty').
'surroundings' indicates putting letters inside.
'o' going inside 'icy' is 'iocy'.
'id'+'iocy'='IDIOCY'
'that's' is the link.
(Other definitions for idiocy that I've seen before include "Complete folly" , "Predatory seabird" , "Foolish action or behaviour" , "Stupid act" , "Foolishness, asininity" .)