  For a non-compact finite thickness building whose Davis apartment is an
orientable pseudomanifold, we compute the supremum of the set of $p>1$ such
that its top dimensional reduced $\ell^p$-cohomology is nonzero. We adapt the
non-vanishing assertion of this result to any finite thickness building using
the Bestvina realization. Using similar techniques, we generalize bounds
obtained by Clais on the conformal dimension of some Gromov-hyperbolic
buildings to any such building.
