Green’s Dictionary of Slang

give a damn v.

also give a dang
[backform. f. not give a damn v.]

to care; usu. in a negative sense, e.g. who gives a damn? or like I give a damn.

[US]W.D. Edmonds Rome Haul 109: Who gives a dang!
[US]J.H. O’Hara Pal Joey 92: Does he give a damn if I am the only one who can operate the radio.
[US]J. Thompson Savage Night (1991) 138: Go ahead and see if I give a dang.
[US]S. Morgan Homeboy 166: [She] could no more pronounce the name of the province pasted on the Army telegram [...] than give a damn.
[UK]Indep. Rev. 9 July 15: It all seems [...] so much harder to give a damn about, than before.