There are several users around who got these kind of "bug" and of course there're several "fixes" out there. I say "fixes" because i wasn't able to find a solution that worked for all users.
There's a reddit about this issue but i'll give you a little summary of all possibilitys the people on reddit posted.
- Clear Cache 5 times straight. A tip from support that helped me solve mine last year
to do so Go to settings, storage and press Y.
- Press the center button on your controller, go to the far right and choose Profile. Then choose Auto Sign-in and disable it. Turn off the xbox and turn it back on. It starts up in seconds now.
Now try to sign in to an account that you don't use that often (if you don't have one, skip this step). Set that one to auto-sign in. Reset xbox, super fast startup...just like the bedroom! Ha!
Next, go to your system>storage>hard drive>profiles and choose the profile that you're having the slow startup with. Notice the profile size (mine was 25MB). Click Delete but choose Delete Profile Only. DO NOT DO PROFILE & ITEMS OR YOU'LL LOSE ALL THE DLC CONNECTED TO THE PROFILE! It'll sign you out if you're signed into that account.
Back on your dashboard, not signed into any profile, press the xbox button on the controller and choose Download Profile. Redownload your profile and start over. Once mine was downloaded again, it was all the sudden 12MB. Half the size of the old one, cool, but ultimately, half the startup time when that profile is set up for auto signin.
- Remove the hard drive, then boot with the hard drive out. Decline the update so it goes to the dashboard. Power off, then insert the drive.
That are all mentioned possible fixes. If you want to know what worked and what not, you've to read the discussion on reddit i've mentioned. I hope any of those will do the trick for you.