How do we know that fundamental constants don't slowly change in time? Let me get one thing straight first. I am not saying that fundamental constants like the speed of light don't have the value that we know they have today. What I am asking is whether the value of a constant could change with time. For example, we measure today that the speed of light is almost $300000000\,m/s$, but this measurement was made over the last century. But what if a long time ago (billions of years ago) that constant had a different value?
So, it's always the limit of how fast something can move, but that limit slowly changes with time.
To conclude,i am asking if there is any kind of evidence that any of the fundamental constants like the speed of light remain constant in time or if we just took that for granted.
 A: We don't know that fundamental constants don't slowly change over time. Au contraire, you can find articles like Changes spotted in fundamental constant: "The researchers found that the fine-structure constant, known as α, has changed in both space and time since the Big Bang".
The thing to note is that some "fundamental constants" aren't constant at all. See the NIST article on the fine-structure constant: "Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is considered an effective or running coupling constant". It's a running constant, which means it isn't constant. 
Thee are other "constants" like this. See http://arxiv.org/abs/0705.4507 and the second paragraph here. Jo Magueijo proposed that the speed of light was faster in the early universe. You can see a mention of that in this Wikipedia article. I'm pretty certain he's wrong myself - it wasn't faster, it was slower. 
PS: the Wikipedia article is wrong about Einstein. He didn't abandon the varying speed of light in 1911. You can find plenty of examples whicih dmeonstrate that in the Einstein digital papers. However in some the English translation used the word velocity rather than speed, causing much confusion. 
