We have measured the specific heat of amber from the Dominican Republic, an ancient geological glass about 20 million years old, in the low-temperature range 0.6 K <= T <= 26 K, in order to assess the effects of its natural stabilization (hyperageing) process on the low-temperature glassy properties, i.e. boson peak and two-level systems. We have also conducted modulated differential scanning calorimetry experiments to characterize the thermodynamic state of our samples. We found that calorimetric curves exhibit a huge ageing signal Delta H approximate to 5 J g(-1) in the first upscan at the glass transition T-g = 389 K, that completely disappears after heating up (rejuvenating) the sample to T = 395 K for 3 h. To independently evaluate the phonon contribution to the specific heat, Brillouin spectroscopy was performed in the temperature range 80 K <= T <= 300 K. An expected increase in the Debye level was observed after rejuvenating the Dominican amber. However, no significant change was observed in the low-temperature specific heat of glassy amber after erasing its thermal history: both its boson peak (i.e., the maximum in the C-p/T-3 representation) and the density of tunnelling two-level systems (i.e., the C-p similar to T contribution at the lowest temperatures) remained essentially the same. Also, a consistent analysis using the soft-potential model of our Cp data and earlier thermal-conductivity data found in the literature further supports our main conclusion, namely, that these glassy 'anomalous' properties at low temperatures remain essentially invariant after strong relaxational processes such as hyperageing.