$40/month just to run it, but then $0.045 per GB data rates. The data rates are what is outrageous. NAT Gateways comprise a non-trivial portion of many customer's bills for this reason.
Exactly, it is $32 just to have it turned on 24/7 and then you pay additionally. Looked into it as it is the only(?) way to get dedicated IP for Lambda which is a common use case, which also explains why it is so costly. They lure you in free tier and then charge for all the necessities.