  We consider the normalized stable tree $\mathcal{T}$ with index $\gamma \in
(1,2]$ and we study the asymptotic behavior of additive functionals of the form
$\mathbf{Z}_{\alpha,\beta}=\int_{\mathcal{T}} \mu(\mathrm{d} x) \int_0^{H(x)}
\sigma_{r,x}^\alpha \mathfrak{h}_{r,x}^\beta\,\mathrm{d} r$as
$\max(\alpha,\beta) \to \infty$, where $\mu$ is the mass measure on
$\mathcal{T}$, $H(x)$ is the height of $x$ and $\sigma_{r,x}$ (resp.
$\mathfrak{h}_{r,x}$) is the mass (resp. height) of the subtree of
$\mathcal{T}$ above level $r$ containing $x$. Such functionals arise as scaling
limits of additive functionals of the size and height on conditioned
Bienaym{\'e}-Galton-Watson trees. We distinguish different regimes depending on
the behavior of $\beta/\alpha^{1-1/\gamma}$ and describe the limiting random
variable in terms of the total height $\mathfrak{h}$ of $\mathcal{T}$ and an
independent $(1-1/\gamma)$-stable subordinator.
