Two-terminal metal/insulator/metal (MIM) memristors exhibiting threshold resistive switching (RS) can develop advanced key tasks in solid-state nano/micro-electronic circuits, such as selectors and integrate-and-fire electronic neurons. MIM-like memristors using multilayer hexagonal boron nitride (h-BN) as dielectric are especially interesting because they have shown threshold RS with ultra-low energy consumption per state transition down to the zeptojoule regime. However, the factors enabling stable threshold RS at such low operation energies are still not fully understood. Here it is shown that the threshold RS in 150 nm × 150 nm Au/Ag/h-BN/Au memristors is especially stable because the temperature in the h-BN stack during operation (i.e., at low currents ≈1 μA) is very low (i.e., ≈310 K), due to the high in-plane thermal conductivity of h-BN and its low thickness. Only when the device is operated at higher currents (i.e., ≈200 μA) the temperatures at the h-BN increase remarkably (i.e., >500 K), which produce a stable non-volatile conductive nanofilament (CNF). This work can bring new insights to understand the performance of 2D materials based RS devices, and help to develop the integration of 2D materials in high-density nanoelectronics.