Cook, who replaced Andrew Strauss, will lead England in four Tests and five one-day internationals in India starting next month.
Broad said: "It's crucial players get behind Alastair and get him off on the right foot.
"Alastair has done really well with the one-day side. I'm sure he'll embrace Strauss' leadership style and add his own style."
Twenty20 skipper Broad also hinted at his interest in the vice-captaincy of the team, the Sun reports.
He said: "It's an important role. I enjoyed being captain at the World Twenty20." (ANI)