The thermal behavior of dilithium-tetrahydridoberyllate Li2BeH4 has been studied up to 825 K by simultaneous DSC-TGA measurements. A solid-solid transition between the hexagonal phenakite form (Be2SiO4) previously described and a possible high-temperature cubic form (Li2SO4-HT) occurs at 536+/-2 K, In addition the thermal decomposition has been characterized by measuring the mass of gas released during heating up to 1075 K. Two distinct steps in the temperature ranges 545-575 K and 905-935 K indicate a total loss of two moles of hydrogen per mole of hydride. The variation of specific heat against temperature has been measured in the 180-420 K range by differential microcalorimetry. A weak but reproducible and reversible anomaly can be detected in the 320-350 K range. This anomaly is marked by a change in the slope of the C-p, vs. T curve. The change in the slope clearly corresponds to a maximum in the vicinity of 340 K on the C-p/T vs T representation, This thermodynamic behavior could be interpreted as a superconductivity criterion when correlated with magnetic irreversibilities previously observed for the same compound, Thus, LiBe2H4 could appear like a type-II superconductor with a critical temperature T-c=(340+/-10) K, However this remarkable potential property requires resistivity measurements to be definitely established,