Lack of interest in a way, desultory in the end (6)
I believe the answer is:
'lack of interest' is the definition.
'a way desultory in the end' is the wordplay.
'way' becomes 'path' (path is a kind of way**).
'in the end' suggests the final letters.
The last letter of 'desultory' is 'y'.
'in' is the link.
(Other definitions for apathy that I've seen before include "Indifference - listlessness" , "Detachment" , "Absense of interest or emotion" , "Impassivity - lack of feeling" , "Lack of interest or enthusiasm" .)