In this paper, we establish the stability result for the $k$-cubic functional equation \[ 2[kf(x+ky)+f(kx-y)]=k(k^2+1)[f(x+y)+f(x-y)]+2(k^4-1)f(y), \] where $k$ is a real number different from $0$ and $1$, in the setting of various $\cal L$-fuzzy normed spaces that in turn generalize a Hyers--Ulam stability result in the framework of classical normed spaces. First we shall prove the stability of $k$-cubic functional equations in the $\cal L$-fuzzy normed space under arbitrary $t$-norm which generalizes previous works. Then we prove the stability of $k$-cubic functional equations in the non-Archimedean $\cal L$-fuzzy normed space. We therefore provide a link among different disciplines: fuzzy set theory, lattice theory, non-Archimedean spaces and mathematical analysis.