From an observer's point of view, the universe grows at the speed of light since the Big Bang if it's flat. It's always finite. The boundary doesn't age, so it's still at the Big Bang with an infinite density.

From a global point of view, a flat universe went from a volume of 0 to an infinite volume instantly at the Big Bang. The universe contains matter that goes from a speed of 0 to infinity, and it can go from a volume of 0 to infinity instantly. The speed between each piece of matter is proportional to its distance.

The speed in the global point of view is equal to gamma*the speed in the observer's point of view. So a speed of c for an observer is equal to a speed of infinity globally as gamma is infinite for v=c.