555 timers consume on the order of single milliamps of current (the lowest is around .5 mA as far as I know). That's far from the lower limit as far as timing-capable ICs go, though, as even a small microcontroller can easily work with consumption around 200 μA and some (if not most) RTCs consume mere nanoamps (not that those will help in this particular case, since you'll require a microcontroller to extract the time from the IC).
Now let's look at the symmetric multivibrator. At any given moment exactly one of the transistors is conducting while the other is not, so there's current going through that transistor's B-E junction and R2/R3(depending on which of the two is conducting), and also through R4/R1 and the C-E junction. Additionally, in your particular schematic there's constant current through R1/R4 and the respective diodes. All in all we have current going through both 470Ω resistors and one 10K resistor (additionally, the capacitor on the opposite side is being charged via the other 10K resistor, but we'll ignore that for now), giving us a current on the order of 9V*(2/470Ω + 1/10KΩ) ≈ 40 mA. This is several times higher than what the ICs can reach.
Note though, that, barring possible minor nonlinear effects, this is mostly dependent on the resistors used; scaling all the resistances up several times dramatically reduces the current consumed by the multivibrator itself, with the only drawback being that the usable output current is reduced accordingly, however in that case you could just use another transistor to switch your load.
Another obvious thing to do would be to move the LEDs to not bypass the transistors, cutting the current used almost in half. Similarly, if you only use one of the halves for your switching, putting a large resistor in place of the load will cut the losses in the idle state.
To summarize: discrete timing circuits, including symmetric multivibrators, can be more efficient than IC-based ones, but they can also be very inefficient, and the particular configuration posted by you is one of the latter.