Area of a loop of a curve in polar coordinates

The following returns the result you expect

Assuming[n > 0 && n ∈ Integers,
 Block[{r = Cos[n θ]^(1/n)},
  FullSimplify[a^2 Integrate[r^2/2, {θ, -Pi/(2 n), Pi/(2 n)}]]]]
(* (a^2 Sqrt[π] Gamma[1/2 + 1/n])/(2 Gamma[1/n]) *)

One way, if symbolically doesn't work is to interpolate:

l = Table[{a, n, 
     1/2 NIntegrate[(Max[0, Power[a^n Cos[n t], 1/n]])^2, {t, 0, 
        2 π/n}]}, {a, 0.1, 5, 0.1}, {n, 0.1, 5, 0.1}] // N;
ip = Interpolation[Flatten[l, 1],InterpolationOrder->5];
ip[1, 3]

0.343417