The board will "stand firm against outside influences," board Chairman Solomon Badger said during a board meeting held by conference call, the Tallahassee Democrat reported.
The state medical examiner ruled last week the Nov. 19 death of a clarinet player drum major, Robert Champion, 26, was a homicide. Officials said he was beaten, and his injuries were consistent with hazing.
Champion was found unresponsive in a parking lot after the team's game against rival Florida school Bethune-Cookman University. Police said Champion vomited and complained he couldn't breathe shortly before he collapsed.
He died at a hospital.
Any death involving hazing is a third-degree felony in Florida.
Longtime university band director Julian White was suspended after the death, and all band performances were canceled.
The Republican governor met with Ammons Friday and stressed his desire that the president of the historically black university step aside.
University alumni gathered on campus Sunday and criticized Scott for trying to oust Ammons.
Badger said Monday he recommended "that until the final report results from these investigations are official, the president's status remain the same."
The state Department of Law Enforcement is investigating possible financial fraud in the band as well as the alleged hazing homicide.
"For the sake of appearances, and to assure the public that these investigations are clearly independent, I believe it would have been in the best interest of Florida A&M University for President Ammons to step aside until all of these investigations are completed," Scott said after the board's decision. "However, we have a process in Florida for the administration of the state university system, and that process has been followed. Like all other Floridians, I will abide by the decision made by the Florida A&M University board of trustees."