A $(3+1)$-dimensional Einstein–Gauss–Bonnet effective description of gravity has been recently formulated as the $D\to 4$ limit of the higher dimensional field equations after the rescaling of the coupling constant. This approach has been recently extended to the four-dimensional Einstein–Lovelock gravity. Although validity of the regularization procedure has not been shown for the general case, but only for a wide class of metrics, the black-hole solution obtained as a result of such a regularization is also an exact solution in the well defined $4D$ Einstein–Gauss–Bonnet theory suggested by Aoki et al. (2005) and in the scalar–tensor effective classical theories. Here we study the eikonal gravitational instability of asymptotically flat, de Sitter and anti-de Sitter black holes in the four dimensional Einstein–Gauss–Bonnet and Einstein–Lovelock theories. We find parametric regions of the eikonal instability for various orders of the Lovelock gravity, values of coupling and cosmological constants, and share the code which allows one to construct the instability region for an arbitrary set of parameters. For the four-dimensional Gauss–Bonnet black holes we obtain the region of stability in analytic form. Unlike the higher dimensional Einstein–Lovelock case, the eikonal instability serves as an effective cut-off of higher curvature Lovelock terms for the $4D$ black holes.

一种 $（3+\mathrm{1个}）$维爱因斯坦-高斯-引擎盖引力的有效描述最近被公式化为 $d\to 4$重新定标耦合常数后，高维场方程的极限。这种方法最近已扩展到爱因斯坦-洛夫洛克的4维重力。尽管没有针对一般情况显示正则化过程的有效性，但仅针对广泛的度量标准，但通过这种正则化获得的黑洞解决方案也是定义明确的精确解决方案$4d$Aoki等人提出的爱因斯坦-高斯-引擎盖理论。（2005）和在标量张量有效的经典理论。在这里，我们研究了二维Einstein-Gauss-Bonnet和Einstein-Lovelock理论中渐近平坦的de Sitter和反de Sitter黑洞的自然引力不稳定性。我们找到了Lovelock引力的各种阶数，耦合值和宇宙常数的真实不稳定的参数区域，并共享了允许为任意一组参数构造不稳定区域的代码。对于四维高斯-贝内特黑洞，我们以解析形式获得了稳定区域。与更高维的爱因斯坦-洛夫洛克案例不同，本征不稳定性可作为高曲率洛夫洛克项的有效截止。$4d$ 黑洞。