Will a pet door let in drafts and raise my energy bill?

It depends heavily on construction. Reviewers consistently report that cheap single-flap doors leak air, while dual- or triple-flap doors with magnetic seals and brush strips perform far better. One r/dogs reviewer specifically reported good cold- and heat-weather performance from a steel-framed three-flap wall door, and automatic motorized panels claim air-tight seals, though they cost much more.