“What’s the difference if we add hot water or cold
water to extinguish fire?”

Sreeda asks:

Answer: The thing to be noted here is that the 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 amount of heat and therefore hot water is better for fire extinguishing purpose 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  vaporisation of hot water.