How hard is it to win an appeal on a child custody case?

I am in the midst of a custody battle with my deadbeat ex. All the facts and the law are in my favor (he is only suing me for custody in retaliation for me suing him for never having payed child support for 5 years), and I am fairly confident I will win. My only worry is that he will appeal the ruling and keep this going for as long as his money holds out. How difficult and expensive is it to win an appeal in a custody case, specifically in TN? Thanks in advance.

Given the state of our justice system, pretty hard.