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Hints Beyond $Λ$CDM from Barrow and Tsallis Holographic Dark Energy with GO cutoff

Barrow and Tsallis Holographic Dark Energy (HDE) are two recent extensions of the standard HDE framework, obtained by introducing generalized entropy corrections through the Barrow and Tsallis formalisms. In this work, we examine the cosmological consequences of Barrow and Tsallis HDE implemented with the Granda-Oliveros (GO) infrared (IR) cutoff. After deriving the modified Friedmann equations within the thermodynamic-gravity conjecture, we study the background evolution in both non-interacting and interacting dark sector scenarios, emphasizing the role of the entropic parameter in shaping late-time dynamics. We then confront the model with state-of-the-art observations, including PantheonPlus and Union3 Type Ia supernovae, Cosmic Chronometers and DESI DR2 BAO measurements. Using Bayesian MCMC methods, we constrain the model parameters and compare the performance of BHDE with that of $Λ$CDM. Our results show that BHDE is compatible with current data and can exhibit a mild statistical preference over the concordance model for certain dataset combinations. Overall, the analysis underscores the relevance of generalized entropy frameworks in late-time cosmology and identifies Barrow-Tsallis holography with the GO cutoff as a competitive alternative to $Λ$CDM.