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Motivated by the doubly special relativity theories and noncommutativespacetime structures, thermodynamical properties of the photon gas in a phasespace with compact spatial momentum space is studied. At the high temperaturelimit, the upper bounds for the internal energy and entropy are obtained whichare determined by the size of the compact spatial momentum space. The maximuminternal energy turns out to be of the order of the Planck energy and theentropy bound is then determined by the factor $\big(V/l_{_{\rm Pl}}^3\big)$through the relevant identification of the size of the momentum space withPlanck scale. The entropy bound is very similar to the case ofBekenstein-Hawking entropy of black holes and suggests that thermodynamics ofblack holes may be deduced from a saturated state in the framework of a fullquantum gravitational statistical mechanics.

Entropy bound for the photon gas in noncommutative spacetime