Faulty Vacuum Power Brake Booster
On modern vehicles the brake system is enhanced by a vacuum operated "booster". This makes the brake pedal softer and more responsive. When the engine is turned off the engine no longer produces vacuum to supply the booster so the pedal gets hard.
easiest way to test vacuum pump is , with engine off, pump brake pedal until pedal goes hard keep pressure on brake pedal and start engine if vacuum pump is good, you will feel the brake pedal go softer so you don't have to use as much effort for pedal to be depressed
Usually when a brake pedal feels more stiff than usual, there is a vacuum leak. The brake lines probably should be bled.
Verify that the vacuum source for the brake booster is working. Might need a new booster.
Power brake booster failure or vacuum supply to booster insufficient.
Faulty Vacuum Power Brake Booster
sounds like you have a leak in your Brake Boost Controler, you should also check to see if there is any leaks in the vacuum lines that connect the engine to the brake booster.
Hardness in the brake pedal can because by low brake fluid. Air in the brake lines can also cause the pedal to push hard.
Either the power brake booster is bad, or it don't have any vacuum going to it. That is the part that the brake master cylinder is bolted to.
You probably have air in the lines or the brake booster is losing vacuum. Have the brakes bleed and have someone put a vacuum gauge on the booster,
The brake pedal vibrates.
There is a vacuum attachment on the power brake booster. If there is a vacuum leak this part could be causing the brakes to be hard, or this part could be the vacuum leak. The steering could also be vacuum assisted occurring from the same source. Are there any other symptoms (i.e. vent/floor/defrost not blowing air or changing properly)?