  This paper concerns with the time periodic viscosity solution problem for a
class of evolutionary contact Hamilton-Jacobi equations with time independent
Hamiltonians on the torus $\mathbb{T}^n$. Under certain suitable assumptions we
show that the equation has a non-trivial $T$-periodic viscosity solution if and
only if $T\in D$, where $D$ is a dense subset of $[0,+\infty)$. Moreover, we
clarify the structure of $D$. As a consequence, we also study the existence of
Bohr almost periodic viscosity solutions.
