Isn't that the Core's Job though? Although-wait a minute!
What if the Frozen Flame is like a lich's phylactery? Lavos certainly has the magic for it, and the Frozen Flame is valuable enough. For those of you who don't play Dungeons and Dragons, lemme explain:
A lich is an undead wizard that binds it's soul to a priceless artifact, creating a phylactery. It can work through this phylactery telepathically in some fiction, or it could be very fragile and require constant protection. In the Frozen Flame's case, let's go with the former.
The Lich does this because as long as the phylactery exists, the Lich is essentially immortal. This could either mean the lich is invulnerable, or it could mean that the lich could be destroyed, but it regenerates back to life. Lavos follows this rule to a degree, as it's past self can use the Frozen Flame to change history and thusly prolong it's lifespan.
This could also explain why the Frozen Flame is sentient and Lavos is basically a wild beast. The Frozen Flame is basically Lavos' Soul.