Question
Does the Gita have any apocrypha?
Answer
Hindu scriptures, including the Mahabharata which contains the Bhagavad Gita, were originally passed down through countless generations via oral tradition. But nowadays the main source we have on them (other than the limited oral tradition that still goes on in some ashrams) is via written manuscripts, and manuscripts had to be copied and recopied as the material they were written on (either bark in North India or palm leaves in South India) was of poor quality. So as it happens, the best manuscripts we have today of the Bhagavad Gita are not the manuscripts of the Mahabharata, but rather manuscripts of commentaries of the Bhagavad Gita, because a commentary needed to quote accurately the verses it was commenting on.
The famous Adi Shankaracharya wrote a commentary (the Bhagavad Gita Bhashya) which quoted 693 of the 700 verses of the Bhagavad Gita, so the "Shankara recension", the version of the Bhagavad Gita quoted in his commentary, as preserved in the South Indian manuscripts, has served as the accepted reference for verses in the Bhagavad Gita - it's the one relied upon whenever people make translations of the Bhagavad Gita or the Mahabarata (since the Mahabharata has to include the Bhagavad Gita).
But in parallel with the South Indian manuscripts of Adi Shankaracharya, there is another commentary on the Bhagavad Gita that has been diligently preserved in manuscripts - the commentary of the Kashmiri Shaivite philosopher Abhinavagupta. He commented on relatively few verses, but some of the verses he quotes are verses that we otherwise don't have from any other source. The version of the Bhagavad Gita quoted in Abhinavagupta's commentary, as passed down in Kashmiri manuscripts, is known as the "Kashmiri recension." In the introduction to this book, Shripad Krishna Belvalkar describes some of the differences between the Shankara and Kashmiri recensions. This mostly has to do with relatively minor grammatical and syntactical changes that lead Belvalkar to conclude that the Kashmiri recension is probably a distortion by later scribes of the original text of the Bhagavad Gita, which in his view would be closer to the Shankara recension.
But the more important difference is the fourteen extra verses unique to the Kashmiri recension. These would be the only thing that could possibly be called by the term "Bhagavad Gita apocrypha". That's really tiny, but you have to consider that the Bhagavad Gita is a 700-verse portion of a 100 thousand-verse epic poem, so there's not much room for apocrypha. In any case, if you want to know what these verses say, they're given in Sanskrit in the appendix of Belvalkar's book. But if you don't know Sanskrit, you can read Swami Lakshmanjoo's brief commentary on the extra verses. (Lakshmanjoo apparently believed that these few extra verses contained some important mystical truths that only the Kashmiri Shaivites were in possession of.)
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