Not true. At least the U.S. Supreme Court has said so (Northwest, Inc. v. Ginsberg, 134 S. Ct. 1422. 1433 (2014))

DOT does have the authority to regulate FFPs if it so desires to do so, as it is related to airline service. But the thing is DOT did not make new rules for FFPs. Instead, DOT relies on its own existing regulations to regulate FFPs. For example, if a FFP is misleading, DOT can go against the program by the usual 49 USC §41712.