Get a new job.

If what you say is true, and you can find work elsewhere, do so. If they're silly enough to throw away 7 hours of productivity over a 6 minute delay, fuck'em.

For a customer-facing position, your happy ass should be in a seat and smiling fifteen minutes before the client shows up.

For development work, though, especially if you aren't coordinating with other people, you should be able to focus on producing quality work quickly.

The time fetishism I've seen is something I mostly see in folks older than their 30s--basically, they don't get that time isn't the big factor it used to be, especially for developers.