I read a few weeks back that this play has now been included in the Shakespeare canon by Arden, so was interested to read it; the full text is online thanks to John W. Kennedy, who I remember with fondness from my past forays onto humanities.lit.authors.shakespeare where the authorship proponents used to hang out (I believe that like all usenet it has now deteriorated beyond recognition).
The play was produced in 1727 by Lewis Theobald, who claimed to have somehow come by a lost manuscript of Shakespeare’s late play Cardenio, based on an episode from Don Quixote. It’s a little odd that there is no character called Cardenio in Double Falshood, the name of the character having been changed to Julio – by Shakespeare, or by Theobald? The manuscript itself has been long lost, believed destroyed in a fire in 1808. So a reasonable doubt has been hanging around the play since 1727.
Myself, though not an expert, I’m reasonably convinced that most of the first half is by Shakespeare – no particularly memorable quotes, but there’s a feeling of the old master keeping his hand in. But I also suspect that Theobald edited it down – the play is much shorter, and the plot less convoluted, than we normally get with Shakespeare. A lot of the second half is clearly Theobald rather than Shakespeare or Fletcher, and the switch to eighteenth-century rather than seventeenth-century idiom is occasionally jarring.
I do wonder if this very uncomfortable theme was part of the reason that the play was lost. The First Folio includes several Shakespeare plays for which there is no contemporary record of performance, whereas it is known that Cardenio had several stage runs in 1614 and after; if Heminge and Condell had wanted to include it, they surely could have tracked it down. On the other hand a couple of the other late plays are also missing, so it may simply be that Heminge and Condell had better access to the earlier archives (or indeed that our records of missing plays are better for the later period).