  Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a
non-negative constant. In this paper we study the asymptotics of
$P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$,
with $\mathcal{T}$ an independent of $X$ non-negative random variable. As an
application, we derive the asymptotics of finite-time ruin probability of
time-changed fractional Brownian motion risk processes.
