(CNN) Bill Cosby is challenging an arbitration decision that mandated he pay nearly $7 million in fees to a law firm that represented him for nine months in 2015 and 2016, documents show.

The final result of the arbitration was filed in January and upheld a $6.7 million bill Quinn Emanuel Urquhart & Sullivan had charged Cosby -- $4.3 million of which had already been paid, $2 million by Cosby and $2.3 million by his insurer, AIG. $2.4 million remains to be paid.

Originally, the law firm was requesting more than $9 million for its total bill. But that amount was trimmed during arbitration.

Cosby wants the arbitration thrown out and a new hearing with a different arbitrator to settle the dispute.

In the documents, filed in a California court on Friday, Cosby says he hired Quinn Emanuel to represent him after "public claims of sexual misconduct against [him] began to snowball at the end of 2014," and that the firm ended up working on 10 different matters representing Cosby. At that point, Cosby had the firm on a retainer.

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