For $c$ not too much greater than $1$, the circle method will yield an asymptotic of size $$E(A)\sim \mathfrak{S}N^{4/c-1.}$$ where $\mathfrak{S}$ is a constant. In fact, for small values of $c$ you can even obtain an asymptotic for the additive energy of the Piatetski-Shapiro primes (see Balog and Friedlander). Letting $S_{A}(\theta)=\mathbb{E}_{n\leq N}1_{A}(n)e(n\theta)$ we may write $$E(A)=N^{3}\int_{0}^{1}|S_{A}(\theta)|^{4}d\theta,$$ or alternatively $$E(A)=\frac{1}{N}\int_{0}^{1}\left|\sum_{n\leq N^{1/c}}e(\lfloor n^{c}\rfloor\theta)\right|^{4}d\theta.$$
Remark: I am not sure for what the value of $c$ the asymptotic breaks down, but it certainly does not hold with $c=2$.