We report here on an experimental investigation of the temporal behavior of transverse mode instabilities into "fully aperiodic large-pitch fibers" (FA-LPFs) operated in high-power continuous-wave laser configuration. To ensure an effective transverse single-mode emission into FA-LPFs, a perfect index matching between the active core and the background cladding materials (Δn=0) is required. The original design of such fibers enables an effective transverse single-mode emission by strengthening the higher-order mode delocalization out of the gain region, even for high heat load levels, consequently leading to the improvement of the beam spatial quality. The study was conducted over fibers of various gain region diameters, from 58 to 100 μm, for a refractive index mismatch Δn of about +8×10-5. The emitted beam is characterized using both M2 measurements and time traces to study the changeover of a stable temporal behavior to an unstable one.