  We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$
on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$,
$m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the
geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate
their fine asymptotics near the extinction time $T$ in terms of a separable
solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$
is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta
V=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly
studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal
V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal
V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is
bounded above and below, near the extinction time $T$, by multiples of
$(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.
