  In this work we study the orbital stability/instability in the energy space
of a specific family of periodic wave solutions of the general
$\phi^{4n}$-model for all $n\in\mathbb{N}$. This family of periodic solutions
are orbiting around the origin in the corresponding phase portrait and, in the
standing case, are related (in a proper sense) with the aperiodic Kink solution
that connect the states $-\tfrac{1}{2}$ with $\tfrac{1}{2}$. In the traveling
case, we prove the orbital instability in the whole energy space for all
$n\in\mathbb{N}$, while in the standing case we prove that, under some
additional parity assumptions, these solutions are orbitally stable for all
$n\in\mathbb{N}$.
