The height is generally defined for formal group laws over rings of characteristic $p$, but one could define the height of $F$ over $\mathbb{Z}$ to be the height of $F\otimes \mathbb{Z}/p$ over $\mathbb{Z}/p$, assuming this height is independent of the prime chosen.
I expect this to be (easily) false, and stated somewhere in a resource on formal group laws, but I can't find it, and this does hold for the (only) examples I am familiar with, the multiplicative and additive formal group laws $F(x,y)=x+y$ and $F(x,y)=x+y+xy$.
Is there an easy counterexample?