Nothing at all that I'm aware of... at first.
However, if this Congressman refused to stop at his (or her) party's behest, assuming they have one (and aren't Independent), they would be subject to whatever disciplinary action that the House decides to take.
The Constitution specifies
Each House [of Congress] may determine the Rules of its proceedings, punish its members for disorderly behavior, and, with the concurrence of two-thirds, expel a member. This report by the Congressional Research Service in 2013 discusses discipline in the House, specifically. Quotes are from there, emphasis is mine:
The House may discipline its Members without the necessity of Senate concurrence. The most
common forms of discipline in the House are now “expulsion,” “censure,” or “reprimand”;
although the House may also discipline its Members in other ways, including fine or monetary
restitution, loss of seniority, and suspension or loss of certain privileges. In addition to such
sanctions imposed by the full House of Representatives, the standing committee in the House
which deals with ethics and official conduct matters, the House Committee on Ethics—formerly
called the Committee on Standards of Official Conduct—is authorized by House Rules to issue a
formal “Letter of Reproval” for misconduct which does not rise to the level of consideration or
sanction by the entire House of Representatives. Additionally, the Committee on Ethics has also
expressed its disapproval of certain conduct in informal letters and communications to Members.
The House may generally discipline its Members for violations of statutory law, including crimes;
for violations of internal congressional rules; or for any conduct which the House of
Representatives finds has reflected discredit upon the institution. Thus, each house of Congress
has disciplined its own Members for conduct which has not necessarily violated any specific rule
or law, but which was found to breach its privileges, demonstrate contempt for the institution, or
reflect discredit on the House or Senate.
Summarizing the rest of the report, in reverse order, a "reprimand" is a resolution
adopted by a vote of the
House with the Member “standing in his place,” or
is merely implemented by the adoption of the
This is effectively a "stop doing that" notice.
A censure would be the next step:
In the House of
Representatives, a “censure” is a formal vote by the majority of Members present and voting on a
resolution disapproving a Member’s conduct, generally with the additional requirement that the
Member stand at the “well” of the House chamber to receive a verbal rebuke and reading of the
censure resolution by the Speaker of the House.
In other words, it's a very public shaming (at least among their peers), although it also doesn't carry any lasting penalty.
Finally, there's expulsion:
In re Chapman, the Supreme Court noted the Senate
expulsion case of Senator William Blount
as supporting the constitutional authority of either
house of Congress to punish a Member for conduct which in the judgment of the body “is
inconsistent with the trust and duty of a member” ... expulsions in the House (and in the
Senate) have traditionally involved conduct which implicated disloyalty to the Union, or the
commission of a crime involving the abuse of one’s office or authority.
While there have only been 5 expulsions from the House in its history (3 in the Civil War, two more recently for corruption), this would solve the problem once and for all.
The Congressman would be able to keep it up only until he frustrated the rest of his peers into making him the 6th person ever expelled from the House.