  In this work we consider the semigroup
$e^{-t\mathcal{A}_{k,\,\ell}^{\gamma}}$ for $\gamma>0$ associated to an
anharmonic oscillator of the form $
\mathcal{A}_{k,\,\ell}=(-\Delta)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers
$\geq 1$. By introducing a suitable H\"ormander metric on the phase-space we
analyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^{\gamma}}$ within the
framework of H\"ormander $S(M,g)$ classes and obtain mapping properties in the
scale of modulation spaces $M^{p,q},\, 0<p,q\leq \infty,$ with respect to an
anharmonic modulation weight. As an application, we apply the obtained bounds
to establish the well-posedness for the nonlinear heat equation associated with
$\mathcal{A}_{k,\,\ell}^{\gamma}$. It is worth noting that the results
presented in this paper are novel, even in the case where $\gamma=1.$
