I guess much depends on how this guy got double stacked in the first place. Did he do something crazy pre flop and get lucky? If so, this might edge towards a call, but we really have to be able to give him JJ or AQ to even consider calling. I think shoving AK like this is a donk move, the only hand that can call that he's not crushed by is QQ, and he's still the dog end of a flip. He gets everything that he dominates to fold. Even with hero calling and paying him off 200bb, I can't see this move being profitable long term. Hero has equal chance of having QQ as he does KK/AA, there's six combos of QQ, and three each of KK and AA facing villain's AK, and this shove is always getting called by KK and AA, sometimes getting called by QQ and AK, and never getting called by any non-donk by anything worse.

All the more reason to fold. Punish his shove by only letting his AK race 200bb deep against KK and AA.