  The Einstein-Lovelock theory contains an infinite series of corrections to
the Einstein term with an increasing power of the curvature. It is well-known
that for large black holes the lowest (Gauss-Bonnet) term is the dominant one,
while for smaller black holes higher curvature corrections become important. We
will show that if one is limited by positive values of the coupling constants,
then the dynamical instability of black holes serves as an effective cut-off of
influence of higher curvature corrections in the 4D Einstein-Lovelock approach:
the higher is the order of the Lovelock term, the smaller is the maximal value
of the coupling constant allowing for stability, so that effectively only a
first few orders can deform the observable values seemingly. For negative
values of coupling constants this is not so, and, despite some suppression of
higher order terms also occurs due to the decreasing threshold values of the
coupling constant, this does not lead to an noticeable opportunity to neglect
higher order corrections. In the case a lot of orders of Lovelock theory are
taken into account, so that the black-hole solution depends on a great number
of coupling constants, we propose a compact description of it in terms of only
two or three parameters encoding all the observable values.
