Adding pergola beams to an existing deck
A 2x6 weighs about 2 lbs. per foot, so each 2x6 will weigh about 2 lbs. x 12’ span = 24 lbs.  

Half the weight will be carried by one end, so the weight on each end will be about 12 - 16 lbs. per foot (if the 2x6’s are 12” o.c.) and there are miscellaneous 2x2 wood strips on top too. 

A 2x6 roof joist support can carry about 150 lbs. per foot, depending on the species and grade. 

So, structurally you do not need to add another 2x6 to support your pergola Beams (2x6’s), but you may want to for appearance. 

See these designs

Remember, the best wood for exterior use and natural finish will be cedar or redwood...but you can always paint pressure treated wood.