  We investigate the heat conductivity $\kappa$ of the Heisenberg spin-1/2
ladder at finite temperature covering the entire range of inter-chain coupling
$J_\perp$, by using several numerical methods and perturbation theory within
the framework of linear response. We unveil that a perturbative prediction
$\kappa \propto J_\perp^{-2}$, based on simple golden-rule arguments and valid
in the strict limit $J_\perp \to 0$, applies to a remarkably wide range of
$J_\perp$, qualitatively and quantitatively. In the large $J_\perp$-limit, we
show power-law scaling of opposite nature, namely, $\kappa \propto J_\perp^2$.
Moreover, we demonstrate the weak and strong coupling regimes to be connected
by a broad minimum, slightly below the isotropic point at $J_\perp =
J_\parallel$. As a function of temperature $T$, this minimum scales as $\kappa
\propto T^{-2}$ down to $T$ on the order of the exchange coupling constant.
These results provide for a comprehensive picture of $\kappa(J_\perp,T)$ of
spin ladders.
