The question you ask is not really the question you seek. The answer to what you asked is boringly simple: "No, a mathematical constant cannot change because mathematical constants are defined to be not changing." While there are some philosophical questions about whether mathematics could indeed be the underpinnings of reality, mathematics is more commonly treated as a thing we created to make sense of the universe. If we decided that π is a constant, it indeed is a constant because we defined it thus.
The more interesting question is the empirical one you are looking at. What if the mathematical relationships that are valid at one moment are sightly invalid in the next. This is a slightly more nuanced question. It's asking "If I hypothetically had a perfectly circular object, and measured the ratio of its circumference to its diameter, would it change over time?"
The limit to this, of course, is our ability to make circular objects. We live in 3-space, not flatland, so maybe a sphere is a better choice:
This beauty is the most spherical object we have ever created. No, not Achim Leistner's balding head -- the silicon sphere in front of him. It's part of Project Avagadro, an effort to fix the mass of the kilogram to something other than a particular lump of platinum-iridium sitting in a vault in France. Liestner is the head optician of the effort, and his spheres are incredibly spherical (a 93.6mm sphere that's 35nm out of spherical -- roughly 370 parts per billion out of spherical!)
So we can see that we're not going to notice a change in a billionths place with anything we can make. We can get about 7 or 8 decimal places at the most. But what if we look bigger?
It turns out that space is big. Really big. So mindbogglingly big that even mathematicians have trouble comprehending how enormously big it is. It's also really old. Really really old. In my opinion, it's not quite as mindbogglingly old as it is mindbogglingly big, but you get the idea. Small changes have... substantial implications.
Consider two objects that are at rest in the same reference frame, with no forces acting on them (you counteracted gravity, somehow). You measure their distance to be exactly 1000mm apart. Don't ask me how you did it. Now go measure their distance apart again. Is it 1000mm? No. It's not. It's actually slightly larger. Why? Because of the expansion of space. If you were to assume that the "space" that we measure in is reality, you would find that that implies that space is growing at a constant rate. If you made your measurements 1 second apart, you would find the second measurement to be
1000.0000000000000022685455mm, give or take a few quintilionths.
What does this imply? Lots of things. For example, energy is not precisely conserved in an expanding space like this. All your energy balances are going to be slightly off. But, if you think about it, you really don't notice these effects. They are easily dominated by other effects. A classic question is "is the expansion of space causing LA and New York to drift apart," to which the answer is no. If you fixed two points in space (one over LA and one over NY at an epoch), those points in space would drift. But these effects are much weaker than the electrostatic forces holding our planet together. LA and NY will stay put... or at least stay put as much as their tectonic plates permit.
So you asked what would happen if some constant were to change slowly in the billionth decimal place or something. Would that make any sense? The answer turns out to be "Yes, it makes sense, and we have it in our reality today!" The expansion of space is a slow change in "fixed constants" that astronomers have to account for. However, practically speaking, it isn't all that important.
Of course, an open question would be what if such tweaked values were actually under the control of some external entity like a deity, with intent. In such cases, it's far less clear whether this would matter or not. It's possible that that billionth digit is all the control this deity needs to shape the universe as they see fit. After all, the universe is a big place. Billions and billions of stars. Rounding errors can add up!