Cook was installed as the new captain following the retirement of Andrew Strauss in August and Broad believes that it is unlikely there will be a big difference in style between the two.
The Twenty20 skipper said: "It was very sad to see Andrew go, he was a fantastic captain and I played a lot of my cricket under him. He's been very inspiring.
"Alastair has played lots of cricket under him as well, so I'm sure he'll embrace Strauss's leadership style and add it to his own style.
"It's a new chapter for English cricket and it's crucial that as players we get behind Alastair and get him off on the right foot."
Cook's first tour as captain will be in India this winter.