A few comments:

In your formula, (pi/4) is raised to the power 4 a b/(a + b)^2. It is
interesting to note that that power can be written nicely in terms of the
geometric and arithmetic means of the semiaxes' lengths, namely, it is the
square of the ratio of those means: Letting gm = sqrt(a b) and
am = (a + b)/2 for brevity, your power can then be written as (gm/am)^2.
Your approximation formula for the perimeter of an ellipse could then be
written as

4 (a + b) (pi/4)^((gm/am)^2)

But the fact that the ratio of means is squared is not crucial to the
formula, and so it is natural to ask whether we could do "better" by
raising gm/am to some power p other than 2.

1)   If our objective is to minimize worst |rel. error|, then, by numerical
methods, it can be determined that p = 2.016861... should be used. We then
obtain |rel. error| < 0.00078 . (Of course, a disadvantage of using
p = 2.016861... is that it's not as easily remembered as 2. But p could, if
desired, be rounded to 2.017 or approximated by the rational number 119/59;
in either case, worst |rel. error| would still be roughly 0.00078 .)

2)   If our objective is to make the formula as accurate as possible for
nearly circular ellipses, then it can be shown that p = 1/(2 ln(4/pi)) =
2.0698... should be used and that, when eccentricity e is small, relative
error = (9 + 2/ln(pi/4))/16384 e^8 (1 + 2 e^2 +...). Of course, over all
eccentricities, worst |rel. error| is not so good, now being roughly 0.0028 . But this value of p gives us another feature which can often be useful:
We have an upper bound on the perimeter

4 (a + b) (pi/4)^((gm/am)^(1/(2 ln(4/pi)))) >= perimeter

with equality only when e = 0 or 1.