She also pointed to a recent article in The National Law Journal that said it was not uncommon for top lawyers at a large law firm to bill clients at a rate of $1,000 an hour. Judge Cote noted that one lawyer at Gibson Dunn, one of the firms representing Apple, billed at a rate of $1,800 an hour.
In an attempt to resolve the fee dispute, she directed Apple and lawyers with the Justice Department to mediate the issue with a United States magistrate.
The company especially took issue with Mr. Bromwich, a former federal prosecutor who has served as court appointed monitoron three previous occasions, with moving too quickly to request interviews and meeting with some of Apple's board members and senior executives. The company pointed to his repeated insistence on interviewing former Vice President Al Gore, an Apple board member. The company argued that board members like Mr. Gore had little involvement with the company's pricing policies.
Apple's lawyers argue that Mr. Bromwich continues to press for the meetings even after he was told that some of the people the monitor wanted to meet with had nothing to do with the antitrust procedures.
(Read more: Apple, Amazon downgraded for 'moral' reasons)
For his part, Mr. Bromwich said in a court filing that Apple was providing him with "far less access" and cooperation than he had received in previous monitoring experiences.
Mr. Bromwich, who has a son who is a news assistant with The New York Times, was not available for comment and did not attend Monday's hearing.
Apple had argued the court filing by Mr. Bromwich should disqualify him from serving as monitor because he displayed a bias by disputing some of Apple's complaints about his activities. But Judge Cote rejected that line of argument.
Earlier in the proceeding, Lawrence Buterman, aJustice Department lawyer, told Judge Cote that Apple's attacks on Mr. Bromwichwere consistent with the company not wanting to work with any monitor.
"They don't want anyone checking their work," Mr.Buterman said.