UC Davis says copper-containing sprays should be used at bud swell. Do you regularly spray for peach-leaf curl? I use Micro-Cop, I believe it's approved for organic farms and is therefore lower in toxicity, and it works quite well against peach leaf curl. I imagine it would help your brown rot problem too. You'd need to spray soon!

Brown Rot:

http://www.ipm.ucdavis.edu/PMG/GARDEN/FRUIT/DISEASE/aprbrownrot.html

Micro-cop:

http://www.groworganic.com/item_PFM315_MicrocopStaStuk_M__Twin_Pack_8_O.html?welcome=T&theses=4015044

Carla in Sac

I would recommend extending the time between watering starting 3 weeks before harvest. For instance if you are watering once a week go 2 weeks. Maybe even stop watering until harvest begins. Mulch the tree real well in the spring this will help the tree to require less frequent watering.
I would also recommend that you thin out the canopy to allow more air movement an a bit more sun into canopy. If you do not summer prune as a rule though be careful not to expose too much of the canopy's limbs at once as this could cause sunburn.
Cracking (which leads to the rot)is a common problem with Flavor King Pluot and following these tips should help lessen the problem.

The water witholding is good advice, withold as soon as they start to color up. In addition spray with lime sulfur when dormant but use copper spray as the buds open because apricots are known to be sensitive to sulfur. (the sulfur is not a problem when tree is totally dormant.