The process of fire extinguishing involves absorption of heat.

Fire extinguishing involves absorption of heat and the absorption of heat in converting hot water to steam is very much more than the heat absorbed in heating cold water to the boiling temperature. Hot water readily converts to steam absorbing enormous amounts of heat and therefore hot water is better for fire extinguishing purposes than cold water.

It is true that cold water absorbs more heat than hot water, say about 1% more than boiling water, but it takes more time than the cooling effect produced by vaporization of hot water.

So, Hot water is quicker than cold water and ice.

But, Boiling water absorbs heat in the form of latent heat for the purpose of changing its liquid state to vapors and latent heat of vaporization is quite high for water. So, boiling water is even better than Hot water for extinguishing the fire.

Hence boiling water can extinguish fire more quickly than hot water, ice or cold water.