Electronic components are soldered to the printed circuit board (PCB)
The printed circuit board is a fibreglass/epoxy construction with thin copper tracks etched onto it.
The components will tend to generate heat, but the board will not generate heat.
As you power up the MPC, the components will start to warm up from ambient temperature to the final operating temperature.
During this heat cycle, there will be stresses on the solder joints between the PCB and components.
After a number of heat cycles the solder joint will fail and some part of the MPC will not work correctly.
Obviously if you leave it on, there are less heat cycles, so less stressing of the solder joints. There are also less switch-on transients, which can cause stressed components to fail with the initial surge of power when powered up.
Downsides are mainly:
economic - higher electricity bills.
environmental - you're not being kind to the planet.
some components have degraded life times at higher temperatures, such as electrolytic capacitors.
backlight on the screen is rated for a certain number of thousand hours of operation, so you are shortening it's operating life.
There are lots of factors which affect your decision on how to use the thing. It depends on the quality of the components used, the type of solder, the robustness of Akai's thermal design (often more of an afterthought in the engineering process) and the ambient temperatures.
I would recommend using the MPC for one session rather than multiple short ones. I think it is probably worth turning it off over night or for long periods when not in use. It's also worth saving things and not expecting the electricity supply to remain on. How gutted would you feel when the lights suddenly go out and you've lost ALL of your stuff when it takes 30 seconds to save?
Finally, a bit of history. Taken from the Wikipedia article on the Colossus computer:
http://en.wikipedia.org/wiki/Colossus_computer"Although machines with many valves were known to have high failure rates, it was recognised that valve failures occurred most frequently with the current surge at power on, so the Colossus machines, once turned on, were never powered down unless they malfunctioned."
As you can see the basic concepts are the same. Although modern components are better than 1940-era valves (tubes for all you Yanks) the basic idea is the same.