Electrical – ON/OFF push button circuit with battery under voltage protection

That's clever... Now let's do some math:

• The datasheet of the LTC1515 indicates, in shutdown mode and with Vin>5V, a 25µA max current. That's something you can't avoid. It is a max spec, and the typical value is not given so let's say it's 5µA.
• Then you have the TPS9837P which is always running, and it uses 6µA. With the efficiency of LTC1515 which seems to top out at 50% for 3.3V output and 6V input (and would be a lot lower for 9V input, I think), that makes ~3µA.

So, at least you have average 8µA consumed by this circuit, without the load itself. This is quite good, actually.

Now, let's say we try to keep it simple: type "nanopower regulator" in google, and you see the LTC3388.

It has less than 1µA at no load. It is simpler to implement than the scheme you propose. It has better efficiency when the load is active. And the best part: It's not more expensive than the LTC1515.

I didn't do much research. There may be more interesting choices than the LTC3388, depending on what you need exactly (LTC is damn expensive, there may be better compromises). But if I were you, I'd try to keep it simple.

Edit Note

I had a deeper look at the LTC1515, because I had a doubt about its efficiency at high input voltages, and I actually realized in your case, it's not even better than a linear regulator (see top of datasheet page 4). It seemed indeed strange to me that it would be more efficient, because there are very few integrated charge pumps that can work as voltage halvers. And those kind of pumps need at least two flying capacitors.

So you'd better take a cheaper linear regulator with very low quiescent current if you don't want the switching buck regulator, because LTC1515 provides no benefits in your case.

But it's not that inefficient actually. Because your circuit spends most time sleeping, and given the sleep consumption of the Atmega328P, the consumption will be dominated by the quiescent current of the main regulator, not really by its efficiency. So a linear regulator may be actually acceptable. Do some math to check it, it depends on the duty cycle of the work/sleep states in your specific application.

I can see a few problems with this circuit.

The first one is about the 7805 voltage regulator.In order to get a stable +5V output,the input voltage has to be at least 7,5V.As a battery is discharged,its outputted voltage drops:

You stated that the circuit is to protect against over-discharge,which means to disconnect when the battery is empty. At some point,the \$V_{in}\$ will become too small for the 7805 to work properly.In other words,the circuit may cut off the power earlier and in an unwanted way,the battery still being able to supply power.You can't really know how is a regulator going to behave if you don't run it within its operating conditions(which can be found in the datasheet).From this point of view,the design is unreliable.

Let's suppose the 7805 supplies the 555 with what's needed as intended and exactly when needed,the relay is turned off.You stated that the timer IC will still continue to draw current(this current is called quiescent current):a minimum of 3mA and a maximum of 6mA.The battery will over-discharge like this.

I believe you should choose another circuit.