Suppose there is a time-dependent Hamiltonian and the Schrodinger equation is solved. $$ i\hbar \partial_t U(t) = H(t) U(t) $$

Now, how easy is it to solve a scaled version of the Hamiltonian (e.g., $H'(t) = 2 H(t)$)? For the time-independent Hamiltonian, we can take $U'(t) = U(2t)$. Does a global scaling of the Hamiltonian cause the evolution to change in a non-trivial way?