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The theories known as doubly special relativity are introduced in order totake into account an observer-independent length scale and the speed of lightin the framework of special relativity. These theories can be generallyformulated on the de Sitter and also recently proposed anti-de Sitter momentumspaces. In the context of these theories, we study the statistical mechanicsand to do this, we consider the natural measure on the corresponding extendedphase space. The invariant measure on the space of distinct microstates isobtained by restriction of the natural measure of the extended phase space tothe physical phase space through the disintegration theorem. Having theinvariant measure, one can study the statistical mechanics in an arbitraryensemble for any doubly special relativity theory. We use the constructed setupto study the statistical properties of four doubly special relativity models.Applying the results to the case of early universe thermodynamics, we show thatone of these models that is defined by the cosmological coordinatization ofanti-de Sitter momentum space, implies a finite total number of microstates.Therefore, without attribution to any ensemble density and quite generally, weobtain entropy and internal energy bounds for the early radiation dominateduniverse. We find that while these results cannot be supported by the standardFriedmann equations, they indeed are in complete agreement with the nonsingulareffective Friedmann equations that arise in the context of loop quantumcosmology.

Early universe thermostatistics in curved momentum spaces