  We argue that the Schwarzschild-de Sitter black hole entropy formula does not
imply that the entanglement spectrum of the vacuum density matrix of de Sitter
space is flat. Specifically, we show that the expectation value of a random
projection operator of dimension $d\gg 1$, on a Hilbert space of dimension
$D\gg d$ and in a density matrix $\rho = e^{-K}$ with strictly positive
spectrum, is $\frac{d}{D}\left(1 + o(\frac{1}{\sqrt{d}})\right)$, independent
of the spectrum of the density matrix. In addition, for a suitable class of
spectra the asymptotic estimates ${\rm Tr} (\rho K) \sim {\rm ln}\ D - o(1)$
and $ {\rm Tr} [\rho (K - \langle K\rangle)^2] = a \langle K \rangle$ are
compatible for any order one constant $a$. We discuss a simple family of matrix
models and projections that can replicate such modular Hamiltonians and the SdS
entropy formula.
