Cold greeting - cold but elegant (4)
I believe the answer is:
chic
'elegant' is the definition.
(thesaurus)
'cold greeting cold' is the wordplay.
'cold' becomes 'c' (eg on taps).
'greeting' becomes 'hi' (hi is a kind of greeting).
'cold' becomes 'c' (eg on taps).
'c'+'hi'+'c'='CHIC'
'but' acts as a link.
(Other definitions for chic that I've seen before include "A la mode" , "Trendy" , "Smart - elegant - fashionable" , "Elegant and stylishly fashionable" , "Stylishness, elegance" .)