What is the meaning of uniform distribution of elements in an array?
Suppose that the array $A$ is chosen in the following way:


Sample $n$ real numbers uniformly from the interval $[0,1]$.
Sort the $n$ samples.


Suppose that you choose a further uniform random variable $x$ from the interval $[0,1]$. The expected running time of interpolation search on $A$ and $x$ is $O(\log\log n)$, where the expectation is over the choices of both $A$ and $x$.

In fact, if I understand Gonnet's thesis correctly, the expectation is still $O(\log\log n)$ even for any fixed $x$. See also Gonnet et al.