What evidence exists that the universe is changing?
There is overwhelming physical evidence that the universe is changing. The debate among astronomers is over whether the universe is expanding, a completely different question.
The universe is changing every second of every day. Every second, somewhere in the universe -- and the Hubble Space Telescope, various space probes, and the dedicated work of thousands of astronomers and mathematicians confirm this -- old stars are dying and new stars are being formed from clouds of dust and gases. The processes by which these developments occur take billions of years, but they are occuring. The universe is filled with more galaxies than we can count, and each galaxy is filled with billions of stars. Planets orbit many of those stars -- as identified by the Kepler space probe, which was just shut down due to mechanical malfunction -- and asteroids fly through space, occasionally crashing into planets and leaving craters, exactly like those we can easily see on our Moon.
That the universe is changing is beyond doubt. Whether it is expanding outward, however, is the subject of debate among some astronomers. Through the use of mathematical calculations involving data derived from observations through powerful telescopes, including the Hubble, by the use of radar, and through the use of other sophisticated instruments, distances between objects in space can be determined with great accuracy. By measuring distances between objects repeatedly over the span of years, movement of those items either towards each other or in opposite directions is determined. Because those distances have been found to grow in many instances, scientists concluded that the universe is expanding outward, which would be consistent with the theory of "the Big Bang" as being the cataclysmic event that created the universe some 13-14 billion years ago.
Whether the universe is expanding or not is debatable; that it is changing is not.