The drug that the government uses to kill Godzilla is described as something that will "disable his internal cooling system". After the drug is used at the end, Godzilla freezes. If the drug was supposed to disable his internal cooling system, it would have overheated him, not frozen him.
But it's a bit more complex than that... In fact, in the movie the drug actually does manage to disable Godzilla's cooling system, but in stead of overheating him this triggers a SCRAM-shutdown (=Safety Control Rods Activation Mechanism) as a kind of involuntarily overreaction-thus freezing him in the procedure.
By freezing himself temporarily, Godzilla is able to survive this potentially critical trauma.