  Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H =
T^\theta$ with fixed $0<\theta \le 1$, $$ \int_T^{T+H}|\zeta(1/2+it)|^{2k} dt
\ll H(\log T)^{k^2(1+O(1/\log_3T))}, $$ where $\log_jT = \log(\log_{j-1}T)$.
The proof is based on the recent method of K. Soundararajan for counting the
occurrence of large values of $\log|\zeta(1/2+it)|$, who proved that $$
\int_0^{T}|\zeta(1/2+it)|^{2k} dt \ll_\epsilon T(\log T)^{k^2+\epsilon}. $$
