Australia reclaimed the Ashes on Tuesday with a 150-run victory in Perth, giving them an unassailable 3-0 lead in the series with two Tests still to play.
The pressure is mounting on Flower following England's inept display with both bat and ball in Australia.
Cook admits the series defeat is the lowest moment he's ever experienced during his 100-Test career.
But he said it was unfair to blame the coach for their horror tour.
"I do want him to carry on. He's an outstanding coach," Cook said.
"He's a great person to have around, especially for me as a captain.
"It's not down to him why we've lost this.
"It's down to the fact we haven't had enough players in form."
Cook vowed not to make any rash decisions in the wake of three consecutive heavy defeats.
But he said the team would embark on some soul-searching in the coming days to ensure they avoid a 5-0 series whitewash.
"There's always going to be inquests when you lose. The simple fact of the matter is we haven't had enough players in form with either bat or ball," Cook said.
"It hurts like hell when you come into a contest and you end up being second best.
"We've had plenty of success and this is the other end of it. It's not pleasant."
Cook averaged 127.7 with the bat during England's 3-1 Ashes triumph in Australia three years ago.
This series he's averaging just 25.7, but he denies the captaincy has become a burden too heavy for him to carry.
"There's always a strain on it," Cook said.
"But I can honestly say that when I go out there as a batter, I'm thinking as a batter.
"My mind is not thinking about what's going on elsewhere.
"It's a tough place, my batting at the moment.
"I'm putting the work in but not quite getting the results.
"You can see the feet aren't always going in the right place."
Cook said his team's poor form wasn't through lack of effort, but rather lack of form.
"I'll never be embarrassed for the way we go about things and the effort the lads put in," Cook said.