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The R\'enyi entropies $R_{p}[\rho]$, $p>0,\neq 1$ of the highly-excitedquantum states of the $D$-dimensional isotropic harmonic oscillator areanalytically determined by use of the strong asymptotics of the orthogonalpolynomials which control the wavefunctions of these states, the Laguerrepolynomials. This Rydberg energetic region is where the transition fromclassical to quantum correspondence takes place. We first realize that theseentropies are closely connected to the entropic moments of thequantum-mechanical probability $\rho_n(\vec{r})$ density of the Rydbergwavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$; so, to the $\mathcal{L}_{p}$-normsof the associated Laguerre polynomials. Then, we determine the asymptotics$n\to\infty$ of these norms by use of modern techniques of approximation theorybased on the strong Laguerre asymptotics. Finally, we determine the dominantterm of the R\'enyi entropies of the Rydberg states explicitly in terms of thehyperquantum numbers ($n,l$), the parameter order $p$ and the universedimensionality $D$ for all possible cases $D\ge 1$. We find that (a) theR\'enyi entropy power decreases monotonically as the order $p$ is increasingand (b) the disequilibrium (closely related to the second order R\'enyientropy), which quantifies the separation of the electron distribution fromequiprobability, has a quasi-Gaussian behavior in terms of $D$.

R\'enyi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics