If you want your fire to get hotter, but don't mind if it stays the same size, you can set the heat release rate per unit area using a ramp.
Another option might be to keep the HRR the same, but enable additional fire vents as the simulation progresses.
Or, if you know a lot about the material you are burning, you could have put your faith in the flame spread model.
It sounds like you want to use a normal, fixed-size, burner with a T2 ramp-up time. Is that correct?