Figaro, a Kingdom in the desert, was Sabin's home, and his responsibility. When Sabin first joins the party and they make their way to Figaro Castle, Sabin feels right at home, taking a seat at one of the twin thrones. He notes that not much has changed, but that Figaro would never be the same after what happened. That something that happened, of course, was his father's death. More so, it was his people's insensitivity toward the matter, and how all they could think about was succession, even just moments after his father's death.
Because of that, I think that Sabin had a love/hate view of Figaro. On one hand, it was his childhood home. He had memories of a better life there with his father and brother. No matter how long he would be away from Figaro, it would always be home to him. It would always be what he knew. On the other hand, Figaro was a place that brought back really dark memories like his father's death, and the selfishness and insensitivity of his people. It would take a lot for Sabin to forget those dark shadows, if he could at all.
I think Sabin stopped caring for Figaro on the night of his father's death. The way the people in his Kingdom were acting made Sabin sick, and he even tried to talk Edgar into abandoning Figaro with him and finding a life elsewhere. Those feelings didn't last long, however. Edgar reminded Sabin that Figaro was their responsibility, and that their father was counting on them to take care of it.
I think the only reason Sabin began to care about Figaro again was because his father wanted him and Edgar to take care of it. Sabin obviously cared about his father, due to how much he broke down after his father's death. He would definitely care about his father's desires for Figaro.
One might argue that Sabin didn't care about Figaro, because he abandoned it for ten years, choosing his own freedom over the Kingdom. I think it proves the opposite, though, especially after watching Sabin and Edgar's ending. In the ending, Sabin reveals that he didn't turn his back on the Kingdom. He wanted what was best for it and he knew that Edgar would be a better King. He also started training, knowing that he'd have to help Figaro one day.
That means the choice for "freedom" that Sabin made wasn't for his own desires. Leaving Figaro, training with Duncan, and staying so close to Figaro were all for Figaro. I think to Sabin, though, they were all for his father.