  Using a double-well potential as a physical memory, we study with experiments
and numerical simulations the energy exchanges during erasure processes, and
model quantitatively the cost of fast operation. Within the stochastic
thermodynamics framework we find the origins of the overhead to Landauer's
Bound required for fast operations: in the overdamped regime this term mainly
comes from the dissipation, while in the underdamped regime it stems from the
heating of the memory. Indeed, the system is thermalized with its environment
at all time during quasi-static protocols, but for fast ones, the inefficient
heat transfer to the thermostat is delayed with respect to the work influx,
resulting in a transient temperature rise. The warming, quantitatively
described by a comprehensive statistical physics description of the erasure
process, is noticeable on both the kinetic and potential energy: they no longer
comply with equipartition. The mean work and heat to erase the information
therefore increase accordingly. They are both bounded by an effective
Landauer's limit $k_B T_{eff} \ln{2}$, where $T_{eff}$ is a weighted average of
the actual temperature of the memory during the process.
