An arduino sitting doing nothing with nothing connected takes about 30mA, this is the same order of magnitude as the self discharge leakage of a battery.
Yes that 30mA is taken at 12V, some is dissipated by the regulator and some by the arduino. A more efficient way of regulating is by a switch mode regulator. These are typically 85% efficient, so a drain of 30mA at 5V becomes a drain of 5 / 12 = 0.416 / 0.85 = 0.489 * 30mA = 14.6mA.
So it is about half the drain but it is still a drain and it is still small.