  Recently, many experiments have been conducted with the goal of demonstrating
a quantum advantage over classical computation. One popular framework for these
experiments is Gaussian Boson Sampling, where quadratic photonic input states
are interfered via a linear optical unitary and subsequently measured in the
Fock basis. In this work, we study the modal entanglement of the output states
in this framework just before the measurement stage. Specifically, we compute
Page curves as measured by various R\'enyi-$\alpha$ entropies, where the Page
curve describes the entanglement between two partitioned groups of output modes
averaged over all linear optical unitaries. We derive these formulas for
$\alpha = 1$ (i.e. the von Neumann entropy), and, more generally, for all
positive integer $\alpha$, in the asymptotic limit of infinite number of modes
and for input states that are composed of single-mode-squeezed-vacuum state
with equal squeezing strength. We then analyze the limiting behaviors when the
squeezing is small and large. Having determined the averages, we then
explicitly calculate the R\'enyi-$\alpha$ variance for integers $\alpha > 1$,
and we are able to show that these entropies are weakly typical.
