How much does volume matter when breaking in amps and cables?

Abery,
The short answer is:
1.  40 hours
2.  Start low and work the volume level up with each run increasing the run time as well
3.  Use a variety of music to cover different frequencies
4.  You can use 8 ohm 100 watt dummy loads in place of speakers
5.  Unless the inputs are couple using a capacitor or transformer you do not need to do each input for 40 hours

BUT 
Break in or Burn in as it was called in the 80's and 90's was done because of part failures.  It was found that if a part was going to fail it would fail within the first 40 hours.  Therefore, products received some run time and samples were run for longer periods.  This was done in a controlled environment with measurement equipment.

Note the word controlled and measurement.  Words like Stress, Cook, Continuously Powered for long periods are not used.

So why do this today? 

Products can fail today.  The forums contain post of fires and major failures in products made in China.  Reviewers have had failures and even fires, see Stereophile.

So what you are trying to do is:
1  Ensure that the product is not going to fail
2  Do so where you can monitor it and make sure it does not take out another piece of equipment
3  Gradually loosen up capacitors, inductors, transformers, and speakers

Do not 
1  Leave the equipment unattended
2  Push the equipment as hard as you can
3  Leave the equipment on all the time as the power has to go somewhere
4  Run it in a 90 degree room
5  Violate manufacturers recommendations
6  Attempt to stress the equipment and therefore create its death
7  Run the equipment when it is too hot to touch.  If this is normal then it is a poor design

Just use common sense.  
1  If the equipment is getting hot to the touch, turn it off.  
2  If it is making noises, turn it off.
3  Follow normal listening practices.  Just because an amplifier can do 100 watts does not mean you should run it for 10 hours straight into 100 watts.
4  Use slow ramp ups in volume and run times.  Many pieces of equipment fail at turn on just like a light bulb

After 40 hours or a period of 5 to 6 days you should be able to feel comfortable with the equipment under normal operating conditions.  And the various bits should be loosened up.

from pure science prospective "breaking in" cables is kinda nonsense....however as all kind of cables age micro cracks are developed in insulation and solder joints . making them worth over the years . cables will never be better over time. i wouldn’t worry about speaker wires ,besides banana or other type of connectors contacts or soldering there is nothing to worry about.
now about "break in" in electronics.electronic components age in time under heating and cooling conditions. resistors increase in value and capacitors decrease.it might be some changes in inductors . when scientific labs build very precise electronic meters they use aging process . they heat and cool components numerous times and then measure them for actual value.The value can change ...i would say under 5%. and then they build the device .So in our case of consumer electronics , components deviate 5-20% of stated value and the hole "enchilada" works just fine. there is no exact formula for "aging" electronic components .i would guesstimate "break in " period from 6 month to a year.would anybody hear the difference in sound? i doubt that very much ... will the system sounds better or worth ? it can go both ways...don’t forget that "break in" in speakers is more "dramatic" and happens only because of mechanical aspect. as semiconductors go, they work or they don’t.
My advice as a specialist in consumer electronics is .... do not use your system for the first 2 weeks  in high volumes, so not to cause damage to speakers.... that's it..... enjoy ! 

Millrcarbon. Funny you mention engine break in 

I managed a plant for Delphi back in the day. The saying was “break it in how you are going to drive it”. If something fails pushing the car a bit hard in the first few miles, it’s not the rpm’s it’s a faulty component. 

It is all quite real. Everything, ime, requires some time to break in, or " settle ", and this is determined by listening, nothing more, nothing less. No measurements, just listening. It is easier to hear, if ( 1 ) you have a system that is highly resolving ( 2 ) a system familiar to you; ( 3) constant, and consistent, with minimal changes in the system; ( 4 ) with music you have heard over and over again; ( 5 ) the ability to listen, with excellent, and sometimes, experienced listening; ( 6 ) , and with, maybe, a trained ear, to hear this break in, warm up, settling in.