  The RG-2 flow is the two-loop approximation for the world-sheet non-linear
sigma model renormalization group flow. The first truncation of the flow is the
well known Ricci flow, at two loops higher order curvature terms appear,
changing almost completely the behaviour of the evolution equation. In this
article we study the RG-2 flow in the context of general relativity. Contrary
to what happens with the Ricci flow the RG-2 flow theory has not been studied
exhaustively, and from a mathematical point of view there are big differences
between both flows. Considering a time symmetric foliation of an asymptotically
flat spacetime, we show that the area $A$ of a closed two dimensional surface
$S$ is monotonous under the RG-2 flow, refining and extending the previous
results already known for the Ricci flow. We show that the inequality that
relates the area of the surface $S$ and the Hawking mass, already found for the
Ricci flow, is still satisfied when we make evolve the area under the new flow.
Finally, we discuss about Perelman`s $W$-entropy for the RG-2 flow and the
physically motivated path towards the gradient formulation of the RG-2 flow.
