  We compute the one-loop quantum corrections to the kink energies of the
sinh-deformed $\phi^{4}$ and $\varphi^{6}$ models in one space and one time
dimensions. These models are constructed from the well-known polynomial
$\phi^{4}$ and $\varphi^{6}$ models by a deformation procedure. We also compute
the vacuum polarization energy to the non-polynomial function
$U(\phi)=\frac{1}{4}(1-\sinh^{2}\phi)^{2}$. This potential approaches the
$\phi^{4}$ model in the limit of small values of the scalar function. These
energies are extracted from scattering data for fluctuations about the kink
solutions. We show that for certain topological sectors with non-equivalent
vacua the kink solutions of the sinh-deformed models are destabilized.
