Teresa worried about university being spartan (7)
I believe the answer is:
austere
'spartan' is the definition.
('austere' can be a synonym of 'spartan')
'teresa worried about university' is the wordplay.
'worried' indicates an anagram.
'about' indicates putting letters inside.
'university' becomes 'u'.
'teresa' with letters rearranged gives 'astere'.
'astere' going around 'u' is 'AUSTERE'.
'being' is the link.
(Other definitions for austere that I've seen before include "Sure eat up and become without luxury" , "Severe in manner" , "Severe; lacking luxuries" , "Harsh" , "Stern, severe" .)