Hi,
Certainly the previous response, $$\phi(x+8)=\phi(x)$$ for all x, is true. However, you asked for the period of the function. Usually when we talk of the period of a function f, we mean the least positive p with f(x + p) = f(x) for all x (sometimes this is called the principal period). For example, the period of sin(x) is $$2\pi$$. Aside: if f is a non-constant continuous function and there is a positive p with f(x + p) = f(x) for all x, then there is a smallest such p; i.e. the period of f exists.
For your specific question, if $$\phi$$ is a non-constant continuous function, the period can be an arbitrarily small positive value!
Example:
Let n be a positive integer, $$a={(2n+1)\pi\over4}$$ and $$\phi(x)=\text{sin}(ax)+5$$. Easily then $$\phi(x)+\phi(x+4)=10$$ for all x, and so this function satisfies your equation. The period of this function is $${2\over 2n+1}$$, which can be made as small as desired by taking n sufficiently large.