The Briton, who has set about improving his own scandal-mired sport's image since he was elected in 2013, told a sports business conference that it was about more than one man.
FIFA president Sepp Blatter, the 79-year-old Swiss who has run the soccer body since 1998, is favourite to win a fifth term in office at a vote in Zurich on May 29.
Asked directly whether, from his own experience, the damaged reputation of a governing body could change with the man at the top staying in place, Cookson replied that it would be difficult.
"I think it is very difficult to change the reputation of an organisation without changing the leadership," he told the conference hosted by the Daily Telegraph.
"However even if you do change the leadership and governance and administration, it's not a given that you are going to change and improve the reputation either.
"You have to walk the walk as well as talk the talk. You have to earn that kind of respect and support and that's a difficult process."
Cookson said in a subsequent clarification that he had been making a general comment based on his own experience at the UCI and had "absolutely no intention to comment on other bodies' matters."
The Briton, who ousted Ireland's Pat McQuaid, faced criticism in his first year in the job as he confronted the sport's doping past and the lingering fallout of the Lance Armstrong scandal.
"People will constantly challenge," said Cookson. "That's my experience of the situation. They will constantly look for behavioural patterns in you that they saw in previous leadership.
"So you've got to try to be true to yourself and true to your commitments."
(Reporting by Alan Baldwin, editing by Justin Palmer)
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