They surpassed the previous record of 263-5, set by Royal Challengers Bangalore in 2013
The 33-year-old, England's leading Test wicket taker, missed the fourth match at Trent Bridge which the hosts won by an innings and 78 runs to take an unassailable 3-1 lead in the series.
"Unfortunately, he's not going to make it," Cook told reporters. "He's gutted about it. He's made really good progress from where he was at Edgbaston but this is a week or so too soon.
"He bowled in the nets yesterday but at about 60-70 percent and that's not how you go into a game. The medical team have been brilliant with him but he's still about a week away."
Mark Wood is likely to keep his place in the team in a four-man pace attack with Stuart Broad, Steven Finn and Ben Stokes.
The fifth Test starts on Thursday.