Let $n, d$ be positive integers. I am interested in the open subset $\mathcal U_{n,d} \subset \mathbb P H^0 ( \mathbb P^n_{\mathbb R}, \mathcal O_{\mathbb P^n_{\mathbb R}}(d))$ corresponding to sections without real zeros. When $d$ is odd and $n$ arbitrary, the set $\mathcal U_{n,d}$ is empty since (homeogeneous) polynomials of odd degree always have (non-trivial) real roots.

Question.Suppose $d$ is even. How many connected components $\mathcal U_{n,d}$ has ? Do we know anything about the Betti numbers of $\mathcal U_{n,d}$ ?

Indeed, motivated by this other question, I am trying to figure out if it makes sense to ask for the number of connected components of the space of polynomial contact distributions of even degree on $\mathbb P^{2n+1}_{\mathbb R}$.