Interrupt (RISING) - triggers twice

pylon:
. . .
According to the datasheet edge detection of external interrupts needs the I/O clock which is not available in power down sleep. Only a level trigger can wake the processor from power down state
. . .

The data sheet for the Atmega328p has an error in this respect. These processors can be woken by any interrupt type. Not true however with the ATTiny series.
Also from https://gammon.com.au/interrupt :

Tip:
I received this confirmation from Atmel that the datasheet is wrong about only LOW interrupts waking the processor.

Quote:

Commented by Manoraj Gnanadhas (Atmel)
2015-01-20 06:23:36 GMT
[Recipients: Nick Gammon]

Hello Nick,

Our design team has confirmed that “Note-3 mentioned under Table 10-1” is a datasheet bug. So you can use any type of interrupt (Rising edge/ Falling edge / Low level / Any logical change) to wake up from sleep mode. Sorry for the inconvenience caused.

Best Regards,
Manoraj Gnanadhas

Thus, all interrupt types will wake the processor.