Hi, not sure what you mean? The high pressure pumps should be running all the time to keep the mill lifted off the bearing. The oil is usually fed back through a cooler to the tank. See example PDF of a SAG mill lube system. What sort of mill? Bearings? Set up? Some need the pump on always, some don't.

I only worked with SAG mills where the high pressure lubrication system is used at startup of the mill.
But once was I worked with a Fuller 18x9 mill we had a heating problem in the bearing that was due to a deformity of the trunnion. We tried everything to lower the temperature, but to make the short history finally had to work with high pressure pumps in constantly. It was the only solution and work well for years until I left that operation. Of course, every half year we bought new pump due it was suffering premature wear.

I am referring to is a scrubber and the use of the pedestal is due to a design flaw, since the high pressure pump should cut out after lifting the mill and leaving the low pressure pump running. We have now controlled the excessive temperature rise by adjusting the alignment of the pedestal. This is at the feed end trunnion bearing.

A scrubber & pedestal? You never said anything about that is your original question...

http://mclanahan.com/wp-content/uploads/Rotary-Scrubbers-Brochure.pdf

You talk about the pedestal use being a design flaw.

Is that because you had to add the pedestal and trunnion bearing as the original setup didn’t work?

You say the pump should cut off but then say the temperature is a function of the pedestal alignment and when aligning it fixed the issue?

Basically the HP pump running won't affect the temperature of the bearing in any way under normal circumstances unless the oil gets hot due to being recirculated without the cooler/tank etc.

Have you contacted your vendor/manufacturer? 

Scrubbers do not have trunnion typical ball mill/SAG mills do.

FOR EVERYONE ELSE READING THIS:  

Ask a detailed question + give all and as much details as possible in the original question. It will save you and those willing to help you.. TIME. You need to ask a question, not a piece by piece riddle.