  Following the classical result of long-time asymptotic convergence towards
the Gaussian kernel that holds true for integrable solutions of the Heat
Equation posed in the Euclidean Space $\mathbb{R}^n$, we examine the question
of long-time behaviour of the Heat Equation in the Hyperbolic Space
$\mathbb{H}^n$, $n>1$, also for integrable solutions. We show that the typical
convergence proof towards the fundamental solution works in the class of
radially symmetric solutions. We also prove the more precise result that says
that this limit behaviour is exactly described by the 1D Euclidean kernel, but
only after correction of a remarkable outward drift with constant speed
produced by the geometry. Finally, we find that such fine convergence results
are false for general nonnegative solutions with integrable initial data.
