In the first part of the talk, based on my recent results, I will present a novel approachto black hole thermodynamics where the Bekenstein-Hawking entropy of Schwarzschild black holesis regarded as a nonextensive Tsallis entropy, and its formal logarithm, the Rényi entropy,is considered for the canonical analysis. In the second part I will present some new, unpublishedresults of an ongoing research, where the problem of cosmological information entropy is consideredfrom the nonextensive point of view.