How do Rényi entropies act under unitary time evolution?

Let $$ rho(t)=Urho U^dagger, $$ just to simplify notation a bit. Now notice that $$ rho(t)^2=Urho U^dagger Urho U^dagger=Urho^2 U^dagger $$ since $U^dagger U=I$. Thus, similarly, $$ rho(t)^n=Urho^n U^dagger. $$ So, take the trace of this, remembering that trace is invariant under permutations: $$ text{Tr}(rho(t)^n)=text{Tr}(Urho^n U^dagger)=text{Tr}(rho^n U^dagger U)=text{Tr}(rho^n). $$ Thus, $$ S_n(rho(t))=S_n(rho). $$

They are invariant under conjugation of unitaries, i.e. under the mapping $rho to U rho U^*$. To see this note that $(U rho U^*)^{alpha} = U rho^alpha U^*$. Then we have $$ begin{aligned} S_alpha(U rho U^*) &= frac{1}{1-alpha} log mathrm{Tr}[(U rho U^*)^{alpha}] &= frac{1}{1-alpha} log mathrm{Tr}[U rho^alpha U^*] &= frac{1}{1-alpha} log mathrm{Tr}[U^* U rho^{alpha}] &= frac{1}{1-alpha} log mathrm{Tr}[rho^alpha] &= S_alpha(rho). end{aligned} $$

On the second line we used the above identity, on the third line we used cyclicity of the trace and on the fourth line we used $U^* U = I$ as $U$ is unitary.