# Bekenstein bound and an infinite universe

I have a question about the Bekenstein bound or the holographic principle.

I make the following assumptions:

1. The universe is infinite
2. The universe is homogeneous

These assumptions are at least considered plausible (on large scale). The Bekenstein bound (or holographic principle) say that the information contained in a ball of radius $R$ is at most $q A$ where $q$ is a proportionality constant and $A$ the area of the sphere. But if the universe is homogeneous, its density is constant and so the information contained in a sphere should be proportional to $R^3$; while the bound is proportional to $R^2$. That seems impossible, a big enough sphere will eventually exceed the bound.

A similar argument seems to imply that some sufficiently large portion of the universe should necessarily be a black hole.

What I am doing wrong here?