I heard some AT fans said Flame Princess's character didn't get develop at all, and let me tell you something she did got her character develop but just alittle. As you seen Hot To The Touch, she was learning what's outside of the world like elements and natures after free from her lantern and the fire kingdom. She was trap inside a lantern and haven't got a clue what's out there from the fire kingdom. She was so carious about natures and elements. And let's not forget what she said before

Hot To The Touch

"Enough. You should not toy with the emotions of a fire elemental. You told me you liked me, and you made my flame grow brighter, but then you put me out, which hurt! You're trying to impede me. That must be your purpose, ...but fire's purpose is to burn."

Flame Princess: I understand now. You're a water elemental.

Finn: What?

Flame Princess: You're my opposite. You create water.

Finn: What, cry? No, I j--

Flame Princess: Yeah, you cry and cry all the time.

Finn: N-No. I-I don't usually do that.

Flame Princess: That's your power.

Ignition Point

"Yeah... It's just the air smells bad from your magic tricks, and now I feel sad. I left all my scented candles at the castle. They'd really cut through the magic stink."

So I think her character is learning about the environment like elements and natures in the earth. I know it looks boring but I think it sounds interesting.