Try again, but this time please do not mix your remarks and questions within the quotation of the printed text.
I'll try to answer anyway.
"Cooling the liquid will raise the pump's discharge pressure. If the discharge control valve is now opened, the pump's discharge pressure will drop back as the flow increases. The discharge pressure drops because the pump develops less feet of head at a higher flow rate.
Forget about pressure when dealing with pumps. Head is the only meaningful term in pump work. Discharge pressure is simply and only the result of pump discharge HEAD x product density at any given time.
When the valve is opened, the pump's HEAD will decrease as flow increases. That is due to the pump's head acting on the system curve and presumedly increasing the energy acting on the system, resulting in an increase in flow through the system. The new pump discharge head will settle at a new point of intersection between the pump's head curve and the system curve. That flowrate may also vary with the change in product density, because any change in product density will most likely change the system curve, especially if the viscosity changed too. The pump curve DID NOT change.
Once the new operation point is established, you can calculate discharge pressure if you like. It will be head * density.
The CHANGE IN DISCHARGE HEAD BETWEEN THE TWO OPERATING POINTS IS NOT DEPENDENT ON THE CHANGE IN PRODUCT DENSITY IN ANY MANNER, OTHER THAN HOW THAT MIGHT HAVE AFFECTED THE SYSTEM CURVE. The statement, "The feet of head of liquid has decreased by the same amount" IS NOT NECESSARILY TRUE. The change in head was caused by the change in flowrate in the system. Any change in PRESSURE will be the result of head change between operating points and the new product density x the new discharge head. DELTA P DOES NOT HAVE TO REMAIN CONSTANT.
"The feet of head has gone down, and the specific gravity has goen up by the same amount." ALSO NOT TRUE.
"Therefore, the horsepower or work per barrel of liquid pumped remains constant." Again NOT TRUE. Power consumed is DENSITY X FLOWRATE X HEAD at any given time. You have varied the flow and the density and the head, none of which MUST be equal to any product of the other two.
Calculate your power required and convert that to electric power, take your voltage and calculate the amps.
The following explaines in practical terms why it is advantagous to forget about pressure, until you need to know it, usually for determining the wall thickness of the pipe you will need to connect to the pump discharge.
If you pump diesel with a constant speed centrifugal pump discharging at a head of 1000 feet and your flowrate is 500 cfs, then switch to gasoline, your flowrate will go up and up as the gasoline pushes the diesel down the pipeline. Your discharge head will stay at the same 1000 feet. Your discharge pressure, if you want to know, will drop, because 1000 feet x gasoline density is less pressure. Your gasoline flowrate will increase due to less viscosity then diesel and the pump will eventually move to the new gasoline flowrate as the diesel clears the pipeline. The final discharge pressure will be the density of gasoline x new final head when gasoline has completely displaced the diesel from the pipeline. Power consumption will be whatever it is. That may be less because you now have the same 1000 feet discharge head x a lighter density product, but flowrate has also changed, probably gone up, as the reduced viscosity of gasoline may have really increased the flow quite a lot, so you can't say for sure that power is now less, remained the same, or gone up.
If you controlled the flowrate of the new lighter density gasoline product to the same flowrate of the previous diesel product, then you could say for sure that power required to pump that flow controlled gasoline decreased from where it was when diesel was being pumped, because flowrate stayed the same, density decreased and head stayed the same.
"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek