Levi Johnston filed a petition for custody last month saying he wants 4-year-old Tripp to be in his mother's and father's lives equally.
The couple had agreed in 2010 that Palin would have primary physical custody and the two would share legal custody, according to Thomas Van Flein, Palin's attorney at that time. Johnston was given visitation and had agreed to pay child support.
Palin's current attorney, John Tiemessen, said that as of Oct. 15, the Child Support Services Division reported that Johnston owed about $66,000 in back support.
Palin and Johnston were thrust into the national spotlight as expectant, unwed teenagers in 2008, when Palin's mother, Sarah Palin, was tapped as the Republican vice presidential candidate.
Johnston and Bristol Palin had an on-off relationship before splitting for good. He has since married and has a daughter.
Bristol Palin has appeared in several reality series, including one for Lifetime that documented her life as a single mom.