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Regularized Einstein-Gauss-Bonnet (EGB) theory of gravity in four dimensionsis a new attempt to include nontrivial contributions of Gauss-Bonnet term. Inthis paper, we make a detailed analysis on possible constraints of the modelparameters of the theory from recent cosmological observations, and sometheoretical constraints as well. Our results show that the theory withvanishing bare cosmological constant, $\Lambda_0$, is ruled out by the currentobservational value of $w_{de}$, and the observations of GW170817 and GRB170817A as well. For nonvanishing bare cosmological constant, instead, ourresults show that the current observation of the speed of GWs measured byGW170817 and GRB 170817A would place a constraints on $\tilde{\alpha}$, adimensionless parameter of the theory, as $-7.78\times10^{-16}\le\tilde{\alpha} \leq 3.33\times10^{-15}$.

Theoretical and observational constraints on regularized 4$D$ Einstein-Gauss-Bonnet gravity