It is sometimes said that the constancy of physical law is an assumption of science. That may have been true once, but constancy has been the subject of a great deal of research. Today, experimental evidence places an upper limit on how much the "constants" could have changed. Broadly, the answer is: at most one percent over the lifetime of the universe.
One nice piece of evidence comes from Supernova 1987a, which was special because it was not very far away. Theory predicts that such a supernova would create about 0.1 solar masses of nickel-56, which is radioactive. Nickel-56 decays with a half-life of 6.1 days into cobalt-56, which in turn decays with a half-life of 77.1 days. Both kinds of decay give off very distinctive gamma rays. Analysis of the gamma rays from SN1987a showed mostly cobalt-56, exactly as predicted. And, the amount of those gamma rays died away with exactly the half-life of cobalt-56. For more details, read:
The Compton Gamma Ray Observatory, Neil Gehrels et al, Scientific American, December 1993, pp.68-77Since SN1987a was 170,000 light years away we were seeing light generated 170,000 years ago. This means that radioactive decay ran at the same speed 170,000 years ago as it does now.
SN1987a Light Curves, P. Whitelock et al., in Proceedings of the Tenth Santa Cruz Workshop in Astronomy and Astrophysics, Springer- Verlag, 1991.
Another evidence is the natural nuclear reactor at Oklo, in Gabon. This reactor was actually just an unusually rich body of radioactive ore. So rich, in fact, that when it was formed, it approached critical mass. Studies of the unusual elements found there indicate that reactors acted the same two billion years ago as they do now. If the fine structure constant had been different by as little as one part in a million, the Oklo measurements should have detected that.
Another evidence is in the light from distant galaxies. When you pass starlight through a prism, you can see spectral lines, which just means that there is an excess (or shortage) of light at specific frequencies. Certain atoms (or molecules or reactions) produce distinctive spectral lines. Modern physics has a solid theory for such things, and we can calculate the frequencies from fundamental constants. Therefore, if we look at a distant galaxy, we can tell if certain fundamental constants are different there. Most of the references below discuss this.
Other methods mentioned in the references:
If you followed that, I said that if a planet has been in the same orbit for a long time, then Uranium-235's radioactive decay rate has been unchanged for that same amount of time. And so on. Physics creates a huge web of connections between astronomy and geology. You may find something debatable about any one of these results. However, it is very hard to argue against a great many independent results, each of which fits into a connected web, and each of which places strong constraints on how fast change could be happening.
It should be noted that Young-Earth Creationists Norman and Setterfield, who argued for a decreasing speed of light, state that the dimensionless fundamental constants have not varied. (Otherwise, life as we know it would have been impossible in the recent historical past.) They were apparently unaware that these two claims are inconsistent with each other.
Clifford W. Will, Was Einstein Right, Basic Books 1986. See Chapter 9, "Is the Gravitational Constant Constant?"
Clifford W. Will, Theory and Experiment in Gravitational Physics, chap. 8.4 and part of 14.3(c)
Cowie & Songaila, "Astrophysical Limits on the Evolution of Dimensionless Physical Constants over Cosmological Time," Astrophysical Journal v.453, p.596 1995
The Oklo Natural Nuclear Reactor: 1, 2, 3
P. Sisterna and H. Vucetich, "Time variation of fundamental constants: Bounds from geophysical and astronomical data," Physical Review D 41 (1990) 1034 and 44 (1991) 3096
E. Richard Cohen in Gravitational Measurements, Fundamental Metrology and Constants, V. De Sabbata and V.N. Melnikov, editors, NATO ASI Series C Vol. 230, Kluwer Academic Publishers, 1988
Barrow, J.D. and Tipler, F.J. The Anthropic Cosmological Principle, Oxford University Press, London 1986
Philosophical Transactions of the Royal Society, London A, 310 (1983) 209-363, was a special issue on "The Constants of Physics."
A.D. Tubbs and A.M. Wolfe, "Evidence for Large-Scale Uniformity of Physical Laws," Astrophysical Journal 236 (1980) L105
W.A. Baum and R. Florentin-Nielsen, Astrophysical Journal 209 (1976) 319.