  Regularized Einstein-Gauss-Bonnet (EGB) theory of gravity in four dimensions
is a new attempt to include nontrivial contributions of Gauss-Bonnet term. In
this paper, we make a detailed analysis on possible constraints of the model
parameters of the theory from recent cosmological observations, and some
theoretical constraints as well. Our results show that the theory with
vanishing bare cosmological constant, $\Lambda_0$, is ruled out by the current
observational value of $w_{de}$, and the observations of GW170817 and GRB
170817A as well. For nonvanishing bare cosmological constant, instead, our
results show that the current observation of the speed of GWs measured by
GW170817 and GRB 170817A would place a constraints on $\tilde{\alpha}$, a
dimensionless parameter of the theory, as $-7.78\times10^{-16}\le
\tilde{\alpha} \leq 3.33\times10^{-15}$.
