If you turn the EDP off (not pull the fire handle), the pump operates in a "depressurized mode". In this mode, would you expect that the pump is not working as hard, so perhaps would run cooler?
Yes, the pump is just spinning with no output pressure or volume.
According to the Maintenance Manual, even with zero fluid output from the pump (to airplane systems), the pump can still generate a fluid pressure of 2900psi (compared to 3030psi at 3750rpm at 37.5gallons/min in pressurised mode). Whether this slight reduction in pressure indicates a % decrease in work, I can't say.
In both cases, whether zero fluid output or 37.5 GPM, the EDP is in the pressurized mode. The reference I'm looking at, NG AMM Part 1 quotes 2850psi at 3750rpm at 37.5 gallons/min and 2950psi to 3075psi at zero flow. Either way, there is a lot more work happening when the output is 37.5GPM.