Equipartition for a homogeneous potential ($⟨A|x{|}^s⟩=k_{B}T/s$)

For a classical canonical coordinate with confining potential energy $U=A|x{|}^s$, $A>0$ and $s>0$, integration by parts in the Boltzmann distribution gives $⟨x{U}^{\prime }(x)⟩=k_{B}T$. Homogeneity then gives $⟨U⟩=k_{B}T/s$. The usual quadratic contribution $k_{B}T/2$ is the special case $s=2$.