Carambola support sleep mode? Or maybe other mode with reduced power consumption.
Thanks

No.
All you can do is turn off unneeded network ports.
There are 5 ethernet ports that can be turned off and the WiFi device.

Apart from that the processor doesn't support any power down/sleep modes.

Whether it is possible to switch on and off the device programmatically?
For example, through some set period of time.

Not without external Hardware. As I said, the processor doesn't have a power down mode. So you can't shut it down without removing the power.
Keeping the RAM alive while removing power from the CPU would probably require some major rewiring.
I don't know if you'd gain bootup speed by copying the RAM contents to some external storage and then reload it on boot...
Both scenarios however would still require an outside influence for controlling that sleep mode.

So an external low power microcontroller (like the MSP430) could run a timer with very low power consumption, powering the Carambola up (and down again) on a schedule.