  We apply the first law of thermodynamics to the apparent horizon of the
universe with the power-law corrected and non-extensive Tsallis entropies
rather than the Bekenstein-Hawking one. We examine the cosmological properties
in the two entropy models by using the CosmoMC package. In particular, the
first numerical study for the cosmological observables with the power-law
corrected entropy is performed. We also show that the neutrino mass sum has a
non-zero central value with a relaxed upper bound in the Tsallis entropy model
comparing with that in the $\Lambda$CDM one.
