No - there is no law that explicitly forbids voting twice in the leadership election of a political party. I've seen some suggestions that it could potentially be prosecuted under section 2 of the Fraud Act 2006, which defines an offence of fraud by false representation when an individual:
- makes a false representation
- knowing that the representation was or might be untrue or misleading
- with intent to make a gain for himself or another, to cause loss to another or to expose another to risk of loss.
I think (though I'm not a lawyer) that such a prosecution would be a bit of a stretch, and I don't know of any precedent which would confirm if voting twice in a private contest would extend to being guilty of this offence.
Schedule 6 of the party's constitution, however, makes it clear that one of the rules of the party is as follows:
No person may cast more than one vote in any ballot in any election for any post within the Party, including any Association or Recognised Organisation.
Given that the threat attached to the ballot stating that voting twice would result in membership being withdrawn - i.e. not a criminal consequence - I think we can say fairly safely that 'offence' in this case referred to this internal party rule.
It's maybe worth pointing out that this sentence which was included on the ballot is not new for this contest - it was present on the party leadership ballots in 2019, as well as the leadership ballots for the Scottish branch of the party in early 2020.