But I've always wondered how we know that the speed of light is the same regardless, that the gravitational constant is constant throughout space and time.
There are actually some scientisst that have proposed a Variable Speed of Light theory. In their theory, the speed of light decreases over time. In other words it was much faster in the early universe. The cool thing about it is, once you make that assumption a lot of other things are suddenly explainable (for example, you don't need inflation in the early universe anymore).
João Magueijo has written a book about it called "Faster Than The Speed of Light, The Story of a Scientific Speculation". There is also a BBC documentary about it called "Einsteins Biggest Blunder".