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We investigate the dark energy phenomenology in an extended parameter spacewhere we allow the curvature density of our universe as a free-to-varyparameter. The inclusion of the curvature density parameter is motivated fromthe recently released observational evidences indicating the closed universemodel at many standard deviations. Here we assume that the dark energyequation-of-state follows the PADE approximation, a generalized parametrizationthat may recover a variety of existing dark energy models. Considering threedistinct PADE parametrizations, labeled as PADE-I, SPADE-I and PADE-II, wefirst constrain the cosmological scenarios driven by them using the jointanalyses of a series of recently available cosmological probes, namely,Pantheon sample of Supernovae Type Ia, baryon acoustic oscillations, big bangnucleosynthesis, Hubble parameter measurements from cosmic chronometers, cosmicmicrowave background distance priors from Planck 2018 and then we include thefuture Gravitational Waves standard sirens (GWSS) data from the Einsteintelescope with the combined analyses of these current cosmological probes. Wefind that the current cosmological probes indicate a very strong evidence of adynamical dark energy at more than 99\% CL in both PADE-I, and PADE-II, but nosignificant evidence for the non-flat universe is found in any of theseparametrizations. Interestingly, when the future GWSS data from the Einsteintelescope are included with the standard cosmological probes an evidence of anon-flat universe is found in all three parametrizations together with a verystrong preference of a dynamical dark energy at more than 99\% CL in bothPADE-I, and PADE-II. Although from the information criteria analysis, namely,AIC, BIC, DIC, the non-flat $\Lambda$-Cold Dark Matter model remains the bestchoice, however, in the light of DIC, PADE parametrizations are stillappealing.

Evidence of dynamical dark energy in a non-flat universe: current and future observations