Saturday's 3-0 loss at Victoria Park was Hartlepool's 15th defeat in their last 21 competitive games and prompted the 47-year-old to resign from his post.
Goals from Steven Rigg, Stephen Elliott and Mark Beck lifted Carlisle above their hosts and ensured that Hartlepool have the worst goal difference in the division at minus 12.
Cooper took the reins at Hartlepool in May 2013 after the club's relegation to League Two and safeguarded their Football League status by three points last season.
"We would like to thank Colin for his efforts during the last eighteen months," chairman Ken Hodcroft told the club's official website.
"We are glad that we gave him his break in management and would like to wish him every success in the future."
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