Free energy and boundary anomalies on $mathbb{S}^atimes mathbb{H}^b$ spaces

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 $mathbb{S}^atimes mathbb{H}^b$, 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 $mathbb{S}^atimes mathbb{H}^b$ 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.