$pi$ is just a number. It doesn't vary with anything.
The ratio of the circumference to the diameter of a circle varies with geometry, so in the general case, this definition doesn't hold. But there are other definitions, such as
$$
pi = int_{-1}^{1}frac{dx}{sqrt{1-x^2}},
$$
or
$$
mathrm{The,smallest,positive,number} , x , mathrm{for,which}, arccos(2x) = 0.
$$
The latter definition can also be expressed a power series, which is independent of geometry.
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