The answer depends on the diffeomorphism.
Let me give two examples, both on the standard torus $\mathbb{R}^2/_{\mathbb{Z}^2}$ with coordinates $x,y$.
(Example 1:) $$\phi(x,y)= (x+ 1/2,y).$$
For this example the cone of metrics which is preserved by this $\phi$ is infinitely dimensional, since any metric $g_{ij}(x,y)$ such that the entries depend on $x$ periodically with period $1/2$ and on $y$ periodically with period $1$ is preserved by it.
(Example 2:) Take irrational $\alpha_1, \alpha_2$ such that the ratio $\alpha_1/\alpha_2$ is also irrational and consider the diffeomorphism
$$\phi(x,y)= (x+\alpha_1, y+ \alpha_2).$$ Then every metric that is preserved by this diffeomorphism has constant entries in the coordinates $x,y$, so the space of metrics is finitely dimensional.
These two examples show the phenomena, and actually give an answer:
If there exists a point such that the orbit w.r.t. the iterations of the diffeomorphism (i.e., the set ${x, \phi(x), \phi(\phi(x)),...}is$\lbrace x, \phi(x), \phi(\phi(x)),...\rbrace$ is dense on the manifold, then the cone of metrics preserved by the diffeomorphism is finite-dimensional.
(One can construct examples such that it has dimension $ 1$)
Otherwise it is infinitely-dimensional.
