This question notes a few families of rational functions whose integrals (from $0$ to $1$) give rational multiples of $\pi$. A fairly straightforward explanation is given there and in the related Math.SE question. The results in the Math.SE post imply that the set of polynomials $p\in\mathbb{Q}[x]$ for which $\int_0^1\frac{p(x)}{x^4+2x^3+2x^2-2x+1}\ dx$ is a rational multiple of pi is a space of dimension two. In general, it's easy to construct polynomials $q(x)$ of degree $2n$ for which the set of polynomials $p$ with $\int_0^1\frac{p(x)}{q(x)}\ dx\in\mathbb{Q}\pi$ has dimension $n$: just sum enough inverse quadratics. Can we do any better than this? Specifically:

Is there a polynomial $q(x)$ of degree $n$ for which the space of polynomials $p(x)$ with $\int_0^1\frac{p(x)}{q(x)}\ dx\in\mathbb{Q}\pi$ has dimension $\gt n/2$?