Is there currently any federal or state laws that punish false rape accusations?
There are no such laws that are specific to rape, but there are general laws about false statements. In every state there is some law against making a false statement to a government official, e.g. Washington RCW 9A.76.175 which says that one who "knowingly makes a false or misleading material statement to a public servant is guilty of a gross misdemeanor". To shift context slightly, if you report to the police that Smith stole your lawn mower when in fact you gave it to him, that is a false statement. However, there would have to be clear proof that you lied in your report, and not that there was a misunderstanding. If Smith stole the mower but the evidence did not support a theft conviction, that does not mean that you can be prosecuted for making a false statement (whereas, if someone has a video of you telling Smith "Here's a mower, which I give to you because I like you", then you could almost be prosecuted for making a false statement, were it not for the fact that the video is illegal in Washington).
Perjury is the other related crime: RCW 9A.72.020 "a materially false statement which he or she knows to be false under an oath required or authorized by law".
About the video of the mower being given away... Washington is an all-party consent state, meaning that you can't just record people, you have to have their permission (everybody's permission). RCW 9.73.050 says that information obtained by illegal recording
shall be inadmissible in any civil or criminal case in all courts of general or limited jurisdiction in this state, except with the permission of the person whose rights have been violated in an action brought for damages under the provisions of RCW 9.73.030 through 9.73.080
which is to say, "unless the person(s) who did not give permission to be recorded now give permission for the evidence to be admitted". Since "you" would be the one making a false statement, "you" would have an interest in suppressing the video, thus "you" could withhold permission for the video to be introduced.