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We discuss a new extended gravity model in ordinary $D=4$ spacetimedimensions, where an additional term in the action involving Gauss-Bonnettopological density is included without the need to couple it to matter fieldsunlike the case of ordinary D=4 Gauss-Bonnet gravity models. Avoiding theGauss-Bonnet density becoming a total derivative is achieved by employing theformalism of metric-independent non-Riemannian spacetime volume-forms. Thenon-Riemannian volume element triggers dynamically the Gauss-Bonnet scalar tobecome an arbitrary integration constant on-shell. We describe in some detailthe class of static spherically symmetric solutions of the above modified D=4Gauss-Bonnet gravity including solutions with deformed (anti)-de Sittergeometries, black holes, domain walls and Kantowski-Sachs-type universes. Somesolutions exhibit physical spacetime singular surfaces not hidden behindhorizons and bordering whole forbidden regions of space. Singularities can beavoided by pairwise matching of two solutions along appropriate domain walls.For a broad class of solutions the corresponding matter source is shown to be aspecial form of nonlinear electrodynamics whose Lagrangian L(F^2) is anon-analytic function of F^2 (the square of Maxwell tensor F_mu_nu), i.e.,L(F^2) is not of Born-Infeld type.

Four-Dimensonal Gauss-Bonnet Gravity Without Gauss-Bonnet Coupling to  Matter - Spherically Symmetric Solutions, Domain Walls and Spacetime  Singularities