Strictness (9)
(Other definitions for austerity that I've seen before include "Severity; lack of comforts" , "Severity of manners or life" , "Self-denial of pleasures" , "Absence of luxuries" , "hardship" .)
'strictness' is the definition.
(I've seen this before)
This is the entire clue.
(Other definitions for rigour that I've seen before include "Hardship" , "Severity, harshness" , "Strict precision, of thought say" , "Severity or stringency" , "Preciseness - severity" .)