We note that results given by Eqs. (97), (98), (101) and (102) are different from their counterparts obtained by the MEMs. The similar result is obtained in the maximum-entropy approach to the superstatistics in Ref. [36] based on the conditional entropy. When we employ, in place of the gamma distribution of Eq. (83), the inverse-gamma distribution proposed in Ref. [46], we obtain for ideal gases, We note that entropies in Eqs. (98), (102) and (105) derived by the maximum-entropy ap- proaches to the superstatistics [36, 37] are additive, which is in contrast with the nonadditive Tsallis entropy defined by Eq. (1). This result is not modified even when we adopt any dis- tribution function of β , because S ( β ) is given by S ( β ) = k$_{B}$Nh ( β ) [see Eqs. (96) and (100)] and then S = k$_{B}$N 〈 h ( β ) 〉 $_{β}$, where h ( β ) depends on β but on N . The superstatistics [30, 31] was introduced to account for the non-Gaussian distribution which is well described by the q -Gaussian ( q -exponential) form originating from the nonadditive Tsallis entropy [1]. We then naturally expect that the entropy obtained by the superstatistics is also nonadditive. It is, however, not the case in the first-principles approaches to superstatistics [36, 37] as mentioned above. Alternatively, we may calculate the energy and entropy by using the partition function Z given by Eqs. (86) or (90). For ideal gases, we obtain E ′ agrees with Eq. (36) or (50) if we read ¯ β = β . Although S ′ is nonadditive, its N dependence is different from S$_{q}$ derived by the MEM in Sec. III. f$_{T}$ ( β ) = ¯ β Γ( n/ 2) ( n ¯ β 2 ) n/ 2 β − n/ $^{2}$−2 e − n ¯ β/ 2 $^{β}$, (103) E = DN 2 ¯ β ( n + 2 n ) , (104) S = k$_{B}$N [( D 2 ) ln ( 2 πm h 2 ¯ β ) + ln σ + ( D 2 + 1 )] − k$_{B}$DN 2 [ ψ ( n 2 + 1 ) − ln ( n 2 )] . (105) E ′ = − ∂Z ∂ ¯ β = DN 2 ¯ β , (106) S ′ = k$_{B}$ (ln Z + ¯ βE ′ ) = k$_{B}$N [( D 2 ) ln ( 2 πm h 2 ¯ β ) + ln σ + ( D 2 + 1 )] + k$_{B}$ [ DN 2 ln ( n 2 ) + ln Γ ( n 2 ) − ln Γ ( DN 2 + n 2 )] . (107) 17