Lord Chief Justice Lord Burnett announced at the end of a hearing at the Court of Appeal in London on Wednesday that the court would do its "level best" to give a decision before the end of July.
Robinson, whose real name is Stephen Christopher Yaxley-Lennon, watched the proceedings via video link from prison.
Robinson was locked up in May after he filmed people involved in a criminal trial and broadcast the footage on social media.
The footage, lasting around an hour, was watched 250,000 times within hours of being posted on Facebook.
Robinson was given 10 months for contempt of court, which he admitted, and a further three months for breaching a previous suspended sentence.