  We prove that the energy over balls of entire, nonconstant, bounded solutions
to the vector Allen-Cahn equation grows faster than $(\ln R)^k R^{n-2}$, for
any $k>0$, as the volume $R^n$ of the ball tends to infinity. This improves the
growth rate of order $R^{n-2}$ that follows from the general weak monotonicity
formula. Moreover, our estimate can be considered as an approximation to the
corresponding rate of order $R^{n-1}$ that is known to hold in the scalar case.
