Let's check the math:

The "theoretical minimum possible amount of energy required to change one bit" is 1.38×10^23 J/K
That's according to wiki - Theoretically, room‑temperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second with only 2.85 trillionths of a watt of power being expended in the memory media. From this, the energy required to count to 2x10^256 is 0.7x10^235 Joules and would take 2x10^247 seconds.

Total energy of the sun - estimated to be 1.3 x 10^44 Joules (as calculated here)

Since the difference in power is 10^203, cooling that theoretical computer would do nothing (but a nice Sci-Fi bit there).

In fact, another wiki says that the total energy of all observable Universe is 4x10^69 J.
For those who is interested the total number of atoms: 10^82.
That's just to show how HUGE 10^256 number is...