Einstein–Hilbert action when GR breaks down at substantially large scales, namely the so-called modified theories of gravity (MOG). In general, the popular scenarios belonging to MOG contain $f(R)$ gravity [@23; @24; @25; @26; @27; @28], scalar–tensor gravity [@29; @30; @31; @32], braneworld gravity [@33; @34; @35], Einstein–aether gravity [@36; @37], and Chern–Simons gravity [@38].

In particular, braneworld gravity is a substantially interesting scenario based on the basic idea that the observational universe could be a (3+1)-dimensional surface (the brane) embedded in a (3+1+d)-dimensional space–time (the bulk), with the Standard Model fields and particles trapped on the brane while gravity can be free to access the bulk. This theory of gravity is inspired by the developments of the $M$ theory [@35; @a2]. To be more precise, the (9+1)-dimensional superstring theories are encompassed by the (10+1)-dimensional $M$ theory, which is widely considered as a prospective route to quantum gravity. Braneworld scenarios have two main features: (i) at low energies, gravity is confined on the brane and GR is naturally recovered; (ii) at high energies, gravity leaks into the bulk, exhibiting behaviors in a high-dimensional way. The gravitational effects of extra dimensions offer attractively testable implications for high-energy astrophysics and cosmology, which result 