MortenA,
Maybe I'm missing the point of the poster's question asking for a "formula to calculate the heat generated by a pump".
The equation determines the heat added to the system by the pump; it is based upon the differences in brake and hydraulic horsepowers. It is also as you say, based upon the specific heat of water (1 BTU/lb per degF). For another fluid, you would have to factor in difference in specific heats.
Maybe the poster was really asking for the total system's heat exchange properties; that's a Thermo deal and I hate Thermo.
I'm also thinking about the PD portion of the question and how the equation would change to adapt from centrifugals.....maybe just the constants?? What say you?