Before a car is started, there is no vacuum to power the brake booster so the brake pedal will be firm. With the car running, the brakes get power and the pedal will feel softer due to the assist of the brake booster.
This is a common behavior to pretty much any car with hydraulic brakes.

 

Interesting explanation.
However, in my experience (last winter) it's the vacuum check valve on the power brake booster.
Either yours has become defective or the rubber grommet the valve sits in is deforming from the cold, and releases vacuum, or should I say allows air into the brake booster. Uncommon failure, but wii millions of cars on the road....
On my car, the check valve + new grommet cost me around $40 iirc.

Symptoms of a Bad or Failing Vacuum Brake Booster Check Valve | YourMechanic Advice

Common signs of a faulty vacuum brake booster check valve include the brake pedal being difficult to engage, feeling spongy, or not working at all.

www.yourmechanic.com

Fix it! Potentially it could go out while on Freeway, you'd have a hard time braking.

 

What @Riick said!
Have you had any working done on the vehicle related to the brake system.
If I recall correctly, when everything is working properly, check valve booster, etc; even if the engine is off, the booster is supposed to store enough 'vacuum' to provide assist for braking 2 or 3 times.
If you don't have assist during cold start, it is possible, as mentioned, the check valve failing, or the booster is not holding vacuum, a mis-adjusted brake push-rod maybe.