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Strictly speaking, yes. However, it's a fairly small effect. From 0 to 100 degrees Celsius, the constant pressure heat capacity of water in joules per gram kelvin is constant to two significant figures. You need to measure to at least three significant figures to see any variation, so for rough "back of the envelope" calculations, treating it as constant doesn't introduce a lot of error.

And if the existing temperature is the boiling point or melting point of water, you can add lots of energy and get no increase in temperature as the energy goes into change of state, instead.

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Q: Does the existing temperature of the water effect the amount of energy required to raise the temperature of the water?
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