Maybe this is a question to naive for the MO community! For a projective smooth variety $X$ defined over a field $F$, and for simplicity let's assume $F$ is a number field. One way to define motivic cohomology is through $K$-theory
\begin{equation}
H^i\_M(X,\mathbb{Q}(j))=K\_{2j-i}(X)^{(j)}\_{\mathbb{Q}}
\end{equation}
where $(j)$ means the eigenspace where Adam operators $\psi^k$ act as multiplication by $k^j$. 


From the construction of higher $K$-theory and Adam operators, for a ring $A$, $K\_l(A)^{(j)}$ only depends on the ring structure of $A$, and it has no dependence on whether we consider $A$ and an $L\_1$-algebra through an embedding $L\_1 \rightarrow A$ or $L\_2$-algebra through $L\_2 \rightarrow A$, where $L\_1$ and $L\_2$ are two different fields. So I guess this definition of motivic cohomology has no dependence on whether we consider $X$ as a variety over $F$ or a variety over a subfield (restriction of scalar), e.g. $\mathbb{Q}$.


On the other hand there are other constuctions of motivic cohomology of $X$, e.g. Bloch’s higher Chow groups and Voevodsky's construction, which indeed use the field $F$ in its constructions. I could not see whether $H^i\_M(X,\mathbb{Q}(j))$ from these constructions has dependence on $X$ being considered as a variety over $F$ or over $\mathbb{Q}$. Anyone who can explain why?



Absoluteness of motivic cohomology and restriction of scalars
By the work of Voevodsky, A^1-homotopy theory, or Cisinski-Déglise, Triangulated categories of mixed motives, we have motivic cohomology of arbitrary schemes (maybe noetherian and finite dimensional, but this is the case here) with integral coefficients. For regular schemes, motivic cohomology with Q-coefficients is isomorphic to the K-theoretic version you mentioned, see Cisinski-Déglise, Introduction, Thm 10, footnote 12. So you get independence of the base field. See also their computation in Example 11.2.3.


Presumably this independence can be checked directly with Voevodsky's definition but I haven't tried to do it.


