As a warm-up in words: The sum of twelve uniform random variables is a classic approximation to a normal distribution. What is the joint pdf for the sum of their cubes and the sum of their fourth powers? This is the $d=12$, $p=3$ case of the asymptotic question that follows.

Let $f_d=f_{p;d}$ be the joint probability density function (pdf) of the pair $(A,B)$ of random variables (r.v.'s) $A:=\sum_1^d X_i^p$ and $B:=\sum_1^d X_i^{2p-2}$, where $p\in(1,\infty)\setminus\{2\}$ and $X_1,\dots,X_d$ are independent r.v.'s uniformly distributed on the interval $(0,1)$.

Can one find the asymptotics of $f_d(a,b)$ for $a\in(0,1)$ and $d\to\infty$, with $p$ fixed and (say) uniformly in $a\in(0,1)$ and $b\in(b_{\min},b_{\max})$, where $b_{\min}:=a^{2-2/p}(1\wedge d^{2/p-1})$ and $b_{\max}:=a^{2-2/p}(1\vee d^{2/p-1})$?

As pointed out in answers here and here, an appropriate asymptotics of $f_d(a,b)$ would be enough to determine the asymptotics of the surface area of a unit $\ell_p^d$-ball for $d\to\infty$ -- an apparently unsolved problem(?).