Are there fields $K$ and $L$ such that $\operatorname{Spec}(K[x])$ and $\operatorname{Spec}(L[x])$ are homeomorphic but $\operatorname{Spec}(K[x,y])$ and $\operatorname{Spec}(L[x,y])$ are not homeomorphic?
(Here $x$ and $y$ are indeterminates.)
EDIT. Note that $\operatorname{Spec}(K[x,y])$ and $\operatorname{Spec}(L[x,y])$ are homeomorphic if and only if they are order-isomorphic.
This is because any closed subset of the spectrum of a noetherian ring is a finite union of subsets of the form $V(\mathfrak p)$ where $\mathfrak p$ is a prime ideal.