Duty to family. It’s in show rather than book canon, but in a History and Lore on Robert’s Rebellion, Stannis says:
“It was the hardest choice I’d ever had to make. My brother or my king. Blood or honor. Aerys ruled by right of all the lords in Westeros…but there are deeper, older laws. The younger brother bows before the elder.”
To me, this fits Stannis’ character perfectly – he uses legal reasoning to argue that precedent limits the oath of obedience to a king.