“I was taken aback by the amount of space in President Obama’s speeches that are devoted to a full frontal assault on Donald J. Trump and his administration,” said Karl Rove, the political strategist for former President George W. Bush. “He spends a considerable amount of his time to get up there and trash Trump.”

Ron Kaufman, who was White House political director for the first President George Bush, said Mr. Obama’s language had been strikingly harsh from one president about another. “If you go back and dig up some of the pretty nasty things President Obama has said, I think you would be a bit surprised,” he said. “He gets away with it because of his style.”

Not since Theodore Roosevelt and Herbert Hoover has a president hit the campaign trail after leaving office to actively take on his successor in quite the way Mr. Obama has. Roosevelt actually mounted a comeback against his handpicked replacement, William Howard Taft, while Hoover castigated Franklin D. Roosevelt’s New Deal program as “despotism” at the Republican convention in 1936.