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When science says "the world is 13,000,000,000 years old", they are not saying "I was there. I have a video. I KNOW". They say "based on the evidence that I have, and based on the laws of nature that I observe, the world could have been created like this.

Or maybe not. Maybe there are other scientific laws involved that we just don't know about. We don't know everything yet, and it's possible all of our theories wouldn't work under different conditions. Remember, in the Rambam's time, everyone was convinced that the science of his day was absolutely true, to the extent that we had to "show" how we agree to science.

Or one can add, even if you say that from the Big Bang on everything was natural, the original yesh mi-ayin, the something from nothing, must have been supernatural. If creation is supernatural, there is no reason to stop the supernatural involment by the Big Bang, why not extend it to the end of creation. It says that when Hashem created the world, it was created fully formed. Meaning, Adam was an adult. If he would come to a doctor and ask "how old am I?" the doctor would say "middle aged". Was he or not?

In other words, it's not a scientific question, as it deals with "did Hashem perform this miracle or that miracle".

Many would answer: "Ahh... But it's a much smaller miracle for Hashem to create a "Big Bang" than create a world. Why make miracles bigger than they are?"

The answer to that this is not a scientific argument. Why Hashem makes a bigger miracle instead of a smaller isn't a scientific question. Maybe philosophical, but not scientific.

So in the end, there is no scientific evidence of the age of the universe. Philosophically one may find it hard to believe why Hashem would create the world looking old, but there are no questions based on science.