  We study the effects of higher-order deformations $\beta_\lambda$
($\lambda=4,6,8,$ and $10$) on the ground state properties of superheavy nuclei
(SHN) near the doubly magic deformed nucleus $^{270}$Hs by using the
multidimensionally-constrained (MDC) relativistic mean-field (RMF) model with
five effective interactions PC-PK1, PK1, NL3$^{*}$, DD-ME2, and PKDD. The
doubly magic properties of $^{270}$Hs are featured by the large energy gaps at
$N=162$ and $Z=108$ in the single-particle spectra. By investigating the
binding energies and single-particle levels of $^{270}$Hs in multidimensional
deformation space, we find that the deformation $\beta_6$ has the greatest
impact on the binding energy among these higher-order deformations and
influences the shell gaps considerably. Similar conclusions hold for other SHN
near $^{270}$Hs. Our calculations demonstrate that the deformation $\beta_6$
must be considered when studying SHN by using MDC-RMF.
