What is this feature?

My reference amp, an integrated Myryad, once turned on reverts the volume to a very low decibel level hence removing the dreaded "did I just blow out my speaker moment?"
All of my other amps do not have this feature. Is there a name for it? Any negatives to it other than an added complication? Thanks. 
Nope, no negatives, and I don't believe there is an official name for it.  It is a nice feature, right next to a time delay when powering up a power amp and getting a huge thump in your subs.
When I turn my linestage off, the motorized pot automatically goes back to full off position.   If one wanted to preserve the precise listening level of the last session, this would be a negative, but, apart from that, I don't see a real problem.  I like the assurance that I will not be accidentally blasting the system.
The Alchemist integrated designed by Tim deParavincini when powered on the volume would go to zero before the relay would trip. Never cared for this feature myself. 
I don't know the name of the feature, but I've encountered it in two different versions. When my Classe CAP-101 powered up, the volume would reset to zero and a 10-second countdown ensued before it could be increased. With the Wyred4Sound STI500-II, if the last volume used before powering down is at or below level "29," the volume starts at that level upon powering up. If the last volume is "30" or more, then the volume starts at level "23," a moderate level. I prefer Wyred's approach but never minded Classe's, thanks to having a remote so I didn't have to stand in front of the amp waiting to set the volume.
I don't have this feature, so my morning often starts out at night-time levels (loud).  If I had this feature I would call it "Whew".  
@edgyhassle....sounds like a great alarm clock you've got there. ;)

Great wake-up tunes:

Strangers When We Meet; The Smithereens
I Wanna Be Sedated; The Ramones
Gotta Get Up; Harry Nilsson
Feets Don't Fail Me Now; Little Feat
Walking In L.A.; Missing Persons

...and I take no responsibility for dents in the ceiling over your bed....*L*

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On my Anthem A/V receiver, I can pre-set startup volume for each input, IIRC.

On my Gato pre, "When the preamplifier enters standby mode again it will remember the selected input channel and volume level. If the volume level is higher than 50, it will be reduced to 50 when the preamplifier exits standby mode."
My ML 585.5 provides for these Volume Control options:

Turn On: This setting lets you set a Speaker output and Line output volume level to which your N0585 will default every time you turn it on.

  • 10.0dB – 60.0dB: Choose a turn-on volume level in 0.1dB increments between 10.0dB and 60.0dB. The factory default setting is 40.0dB.

  • Last: The volume level setting from the previous time the unit was powered down is retained.

    CAUTION: Setting the Turn On parameter to Last can result in louder-than-expected power-up volume if the N0585 was set to a high volume level setting when last powered down.

I am not sure there is a name for it but i always turn my amplifier all the way down before powering down because that is how i was taught. It is a very nice feature to have though so you have no accidents while listening.
The name should be-"Hey dumb sh-t I just saved your ears and your speakers". It is on my Pass INT25 and I appreciate it. Thanks.
My ARC preamp has a mute switch and my Cheap A**ed Denon has a user preset for the volume. I set it once and never needed to touch it again.