  In a spatially infinite and eternal universe approaching ultimately a de
Sitter (or quasi-de Sitter) regime, structure can form by thermal fluctuations
as such a space is thermal. The models of Dark Energy invoking holographic
principle fit naturally into such a category, and spontaneous formation of
isolated brains in otherwise empty space seems the most perplexing, creating
the paradox of Boltzmann Brains (BB). It is thus appropriate to ask if such
models can be made free from domination by Boltzmann Brains. Here we consider
only the simplest model, but adopt both the local and the global viewpoint in
the description of the Universe. In the former case, we find that if a
parameter $c$, which modulates the Dark Energy density, lies outside the
exponentially narrow strip around the most natural $c = 1$ line, the theory is
rendered BB-safe. In the later case, the bound on $c$ is exponentially
stronger, and seemingly at odds with those bounds on $c$ obtained from various
observational tests.
