Brake pedal

You're expecting the opposite of what should happen. Unless you have one of a few specific models of anti-lock brake systems, normal power brakes work on engine vacuum, but that is only developed when the engine is running. The power booster encases a large vacuum reservoir. Among its jobs, one is to use that reserve vacuum if the engine stalls while you're driving. At best that will provide two to three pedal applications before the reserve vacuum is used up. You can verify that when you stop and turn the engine off. Wait a few seconds, then press and release the brake pedal. You'll feel it get harder to push each time. That is normal.

Normal operation is to start the engine, then apply the brake pedal if your car needs that to shift out of "park". If you have to push the brake pedal to start the engine with a push button, the best suggestion is to be sure to not push that pedal after stopping the engine last time, so it doesn't exhaust that reserve vacuum.

There is also a check valve in the hose going to the power booster. If that is leaking, the reserve vacuum will be lost too. It's rare for them to leak very quickly, but if they do, the symptom is that reserve vacuum will be lost within a minute or less. After stopping the engine, wait a minute or two, then see if the pedal is already hard to push. If it is, have the dealer look at it. This would be a warranty issue. Just about all check valves are going to leak rather slowly, so it's normal to not have power assist the next morning until the engine is running.