This one is pretty simpler than it sounds: It's actually never explained; your total time bonus is the "Cumulative time bonus" you get from hitting/killing enemies. You'll see a number of seconds pop up on screen based on your multipliers when you hit/kill enemies... that total time has to reach 10 minutes by the end of the round. There is no way to tell in-game what your total is.
Go to "Main House 2" trial and do a loop around the main house killing enemies. The monsters respawn, so if you exit a room and go back, the monster should respawn. Just go around the house and kill monsters. I had 6 minutes left of the trial, and when I gave Jack the final piece of food and finished the trial, I had a 25-minute time bonus.