Dielectric elastomers such as 3M VHB4910 acrylate film have been widely used for electromechanical energy conversion such as actuators, sensors and generators, due to their lightweight, high efficiency, low cost and high energy density. Mechanical and electric properties of such materials have been deeply investigated according to various parameters (temperature, frequency, pre-stress, nature of the compliant electrodes…). Models integrating analytic laws deduced from experiments increase their accuracy. Nevertheless, leakage current and electrical breakdown reduce the efficiency and the lifetime of devices made with these polymers. These two major phenomena are not deeply investigated in the literature. Thus, this paper describes the current-voltage characteristics of acrylate 3M VHB4910 and investigates the stability of the current under high electric field (kV) for various temperatures (from 20°C to 80°C) and over short (300 s) and long (12h) periods. Experimental results show that, with gold electrodes at ambient temperature, the current decreases with time to a stable value corresponding to the conduction current. This decrease occurs during 6 hours, whereas in the literature values of current at short time (less than 1 hour) are generally reported. This decrease can be explained by relaxations mechanisms in the polymer. Schottky emission and Poole-Frenkel emission are both evaluated to explain the leakage current. It emerges from this study that the Schottky effect constitutes the main mechanism of electric current in the 3M VHB4910. For high temperatures, the steady state is reached quickly. To end, first results on the leakage current changes for pre-stretch VHB4910 complete this study.