  For simple graphs $G$, $G_1$ and $G_2$, we denote their double corona based
on $R$-graph by $G^{(R)}\otimes{\{G_1,G_2\}}$. This paper determines the
normalized Laplacian spectrum of $G^{(R)}\otimes{\{G_1,G_2\}}$ in terms of
these of $G$, $G_1$ and $G_2$ whenever $G$, $G_1$ and $G_2$ are regular. The
obtained result reduces to the normalized Laplacian spectra of the $R$-vertex
corona $G^{(R)}\odot{G_1}$ and $R$-edge corona $G^{(R)}\circleddash{G_2}$ by
choosing $G_2$ or $G_1$ as a null-graph, respectively. Finally, applying the
results of the paper, we construct infinitely many pairs of normalized
Laplacian cospectral graphs.
