Could PI have a different value in a different universe?

Physically, the ratio of a circle's circumference to its diameter $C/d$ is not really $\pi$. General relativity describes gravity in terms of the curvature of spacetime, and roughly speaking, if you take $(C/d-\pi)/A$, where $A$ is the circle's area, what you get is a measure of curvature called the Ricci scalar.

But even if you're doing general relativity, you don't just go around redefining $\pi$. The thing is, $\pi$ occurs in all kinds of contexts, not just as $C/d$. For instance, you could define $\pi$ as $4-4/3+4/5-4/7+\ldots$, which has nothing to do with the curvature of space.

So if you define $\pi$ as $C/d$, you don't even get a consistent value within our own universe, whereas if you define it as $4-4/3+4/5-4/7+\ldots$, you get an answer that is guaranteed to be the same in any other universe.

Another way of looking at it is that $\pi$ is not the $C/d$ ratio of a physical circle, it's the $C/d$ ratio of a mathematically idealized circle that exists in certain axiomatic systems, such as Euclidean geometry. Viewed this way, it doesn't matter that our universe isn't actually Euclidean.

This is a complement of other answers. One can define a value $\pi_p$ as $\pi$ in $ \ell_p$. $\ell_p$ is two dimensional space with a metric as follows: $$ d\left((x_1,y_1),(x_2,y_2)\right)=\left(|x_1-x_2|^p+|y_1-y_2|^p\right)^{1/p} $$ for $1\leq p \leq \infty$. Then the circle $C_p$ is defined as all points $(x,y)$ such that: $$ \left(|x|^p+|y|^p\right)^{1/p}=1 $$ The diameter of this circle is $2$. And therefore we can define $\pi_p$ as the half of circumference of the circle. It can be seen that $\pi_1=4$, $\pi_2=\pi$ and $\pi_\infty=4$. In following image, you can see the value $\pi_p$ versus p:

Reference: Look at the following article $\pi _{p} $ the Value of π in $\ell _p $.

That there will be a number $\pi$ is a mathematical fact. But whether the significant number would be the same is a more interesting question. Some people in our own universe would prefer that the constant had been chosen to be $2\pi$ i.e. $6.28 ...$ instead of $3.14 ... $ as it would reduce the number of factors of 2 in some formulae.

It would also be possible to imagine, in a higher dimensional universe, that the basic round object might be, say, a 3-sphere, with the significant constant would be defined in relation to its geometry rather than the geometry of a circle.

Living in a world which was non-euclidian (e.g. on the surface of a sphere) would make other numbers geometrically significant, but there would still be $\pi$ = $3.14 ...$ sitting in the background.