The England captain was unbeaten on 237 as England reached 468 for five at tea, just 55 runs behind Pakistan’s first innings total of 523 for eight declared as the prospects of a result on a flat Sheikh Zayed Stadium diminished.
Ben Stokes was the other unbeaten batsman with 16.
Cook came to the crease in the last hour of the second day and has since batted with authority and grit, smashing 17 boundaries during his resolute 471-ball knock.
He added 141 runs for the fourth wicket with Joe Root who made a brilliant 85 after England resumed at 290-3.
Paceman Rahat Ali provided Pakistan the breakthrough after they went without wicket in the extended two-and-a-half hour session due to Friday prayers.
After lunch Root played a rash shot off a wide delivery and was caught by wicket-keeper Sarfraz Ahmed. He hit seven boundaries during his 143-ball stay.
But it was Cook’s third double century that enlivened an otherwise dull day.
Cook flicked paceman Wahab Riaz to backward square-leg for two to complete his double century off 395 deliveries, the second of the match behind Pakistan’s Shoaib Malik’s 245.
Cook has so far hit 16 boundaries in a 407-ball stay.
Root also batted with authority, reaching his 14th Test half-century with a reverse sweep boundary off Malik.
Pakistan were once again sloppy in the field as they missed another chance to dismiss Cook when wicket-keeper Sarfraz Ahmed failed to take an inside edge off Riaz when the England captain was on 173.
Fawad Alam had let off Cook on 147 off spinner Zulfiqar Babar on Thursday.
Root also survived a confident leg-before shout when fast bowler Imran Khan trapped him but Pakistan’s review once again went unsuccessful.
Riaz was the pick of the bowlers with 3-116.
Pakistan’s frontline spinner Babar also struggled to take a wicket, conceding 132 runs in his 58 overs.