The components of the stress-energy tensor are different in different reference frames. Also there is no universal time, so values of energy will be different in different reference frames.
Via the Einstein field equations, does this not imply that the curvature of spacetime is different depending on your reference frame?
The components of a tensor are dependent on what coordinate or reference frame you use. So yes, the components of the stress-energy tensor and the components of the Riemann curvature tensor change between coordinate systems. However, the tensors themselves are invariant under any and all coordinate transformations.
You can think of the components of a tensor as that tensor's projection on some coordinate basis. These projections are going to change depending on your coordinate system, but the geometric object which they represent does not. For example, the components of the stress-energy tensor are $T^{\alpha\beta}$ for some observer $\scr{O}$ and $T^{\bar{\mu}\bar{\nu}}$ for another observer $\bar{\scr{O}}$. These observers are going to disagree on the components of the tensor $(T^{\alpha\beta} \neq T^{\bar{\alpha}\bar{\beta}})$, but they will agree when they reconstruct the $2 \choose 0$ tensor, $T$
$$ T = T^{\alpha\beta}\vec{e}_\alpha\otimes\vec{e}_\beta = T^{\bar{\mu}\bar{\nu}}\vec{e}_\bar{\mu}\otimes\vec{e}_\bar{\nu} $$
So long each observer uses the basis vectors $\vec{e}$ associated to their own coordinate system.
