Rest live outside university (7)
I believe the answer is:
residue
'rest' is the definition.
('residue' can be a synonym of 'rest')
'live outside university' is the wordplay.
'live' becomes 'reside' (residing is a kind of living).
'outside' indicates putting letters inside.
'university' becomes 'u'.
'reside' going around 'u' is 'RESIDUE'.
(Other definitions for residue that I've seen before include "Matter left behind after other parts have been taken away" , "It remains after other parts have been taken away" , "Material left" , "Stuff left over" , "What is left, maybe dregs" .)