Terribly solemn pair might be frigid (10)
I believe the answer is:
impersonal
'frigid' is the definition.
(thesaurus)
'terribly solemn pair' is the wordplay.
'terribly' is an anagram indicator.
'solemn'+'pair'='solemnpair'
'solemnpair' is an anagram of 'IMPERSONAL'.
'might be' is the link.
(Other definitions for impersonal that I've seen before include "Aloof" , "Cold" , "Objective; disinterested" , "Cool" , "Ignoring one's own feelings" .)