The law of $\mathbf{X}$ is characterized by $\alpha$ and its *cumulant function* $\kappa:\mathbb{R}{\left(-\infty\,,\infty\right]}$, both depending on the driving process.

In this paper we are interested in the cases where $\alpha<0$ and $\kappa$ has two positive root $\omega_-<\omega_+$. Under some further conditions on $\alpha$ and $\kappa$, the growth-fragmentation yields a natural measure on the genealogical tree of the branching process seen as a metric space, namely the *intrinsic area measure*. Denoting $A(t)$ the intrinsic area of the ball of radius $t$ centered at the root of the tree, one can investigate the regularity of the Stieltjes measure $\mathrm{d}A(t)$. This has been studied in [@G18], in which Theorem 1 shows that if $\alpha>-\omega_-$, then $\mathrm{d}A(t)$ is absolutely continuous whereas it is singular when $\alpha\leq -\omega_-$. In this paper, we consider the absolutely continuous case. A noteworthy fact is that the density is null at $t=0$, meaning that the dissipation of the area occurs as small particles are about to vanish. We study further properties of $A$.

Our first result is Theorem \[Theorem area GF\], that determines the almost sure behaviour of $A(\epsilon)$ as $\epsilon\to 0+$, when $\mathbf{X}$ starts from a single particle of positive size $x$. Namely, there exists a regularly varying 