  We compute free energies as well as conformal anomalies associated with
boundaries for a conformal free scalar field. To that matter, we introduce the
family of spaces of the form $\mathbb{S}^a\times \mathbb{H}^b$, which are
conformally related to $\mathbb{S}^{a+b}$. For the case of $a=1$, related to
the entanglement entropy across $\mathbb{S}^{b-1}$, we provide some new
explicit computations of entanglement entropies at weak coupling. We then
compute the free energy for spaces $\mathbb{S}^a\times \mathbb{H}^b$ for
different values of $a$ and $b$. For spaces $\mathbb{S}^{2n+1}\times
\mathbb{H}^{2k}$ we find an exact match with the free energy on
$\mathbb{S}^{2n+2k+1}$. For $\mathbb{H}^{2k+1}$ and $\mathbb{S}^{3}\times
\mathbb{H}^{3}$ we find conformal anomalies originating from boundary terms.
  We also compute the free energy for strongly coupled theories through
holography, obtaining similar results.
