= Solution
Write $r=\phi_0^2/M_{\rm Pl}^2$. As $t\to0$,
$$
\tanh(\lambda t)\sim\lambda t,\qquad
\sinh(2\lambda t)\sim2\lambda t,
$$
so $\phi\to-\infty$ and
$$
a(t)\sim \text{constant}\,t^{r/2}.
$$
Because $2/3<r<2$, the power $r/2$ lies in $(1/3,1)$; thus $\ddot a<0$ and the early expansion decelerates. As $t\to\infty$, $\phi\to0$ and $\sinh(2\lambda t)\sim e^{2\lambda t}/2$, giving
$$
a(t)\sim \text{constant}\,e^{\lambda rt},
$$
so the late universe has <accelerating expansion>.
The transition satisfies $\ddot a/a=H^2+\dot H=0$. Substitution gives
$$
r^2\coth^2(2\lambda t)-2r\operatorname{csch}^2(2\lambda t)=0,
$$
or $r\cosh^2(2\lambda t)=2$. Therefore
$$
\boxed{\
t_{\rm acc}
=\frac{1}{2\lambda}
\operatorname{arcosh}\left(\sqrt{\frac{2M_{\rm Pl}^2}{\phi_0^2}}\right)\
}.
$$
This is the transition described by the <exact hyperbolic-sine scalar-field cosmology>.
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