Abstract Let 𝑀 be a smooth, compact manifold and let N μ \mathcal{N}_{\mu} denote the set of Riemannian metrics on 𝑀 with smooth volume density 𝜇. For g 0 ∈ N μ g_{0}\in{\mathcal{N}_{\mu}} , we show that if dim ( M ) ≥ 5 \dim(M)\geq 5 , then there exists an open and dense subset Y g 0 ⊂ T g 0 N μ {\mathcal{Y}_{g_{0}}}\subset T_{g_{0}}{\mathcal{N}_{\mu}} (in the C ∞ C^{\infty} topology) so that, for each h ∈ Y g 0 h\in{\mathcal{Y}_{g_{0}}} , the ( N μ , L 2 ) ({\mathcal{N}_{\mu}},L^{2}) Ebin geodesic γ h ( t ) \gamma_{h}(t) with γ h ( 0 ) = g 0 \gamma_{h}(0)=g_{0} and γ h ′ ( 0 ) = h \gamma_{h}^{\prime}(0)=h satisfies lim t → ∞ R ( γ h ( t ) ) = − ∞ \lim_{t\to\infty}R(\gamma_{h}(t))=-\infty , uniformly, where R : N μ → C ∞ ( M ) R\colon\mathcal{N}_{\mu}\to C^{\infty}(M) is the scalar curvature.