  We investigate the effects of the Einstein cubic gravity (ECG) on regular
black hole solutions driven by nonlinear electrodynamics (NLE) sources. The ECG
tends to form a naked singularity at the origin for a high ECG coupling
constant. Assuming that ECG provides only perturbative corrections to the
regular magnetic charged solutions, we found modified regular solutions with a
de Sitter-like core whose cosmological constant depends on the magnetic charge
and the ECG coupling constant. The thermodynamic stability is investigated by
means of the Hawking temperature and the heat capacity. In fact, for a small
charge and ECG coupling, the Hawking temperature is regularized, leaving a
thermodynamic stable remnant for a small $r_h \neq 0$. The heat capacity
reveals that the ECG regular black hole undergoes a phase transition between an
unstable into a stable configuration.
