We investigate the L_p-spectrum of linear operators defined consistently on L_p(Ω) for p0 ≤ p ≤ p1, where (Ω,µ) is an arbitrary σ-finite measure space and 1 ≤ Po ≤ p1 ≤ ∞. We prove p-independence of the L_p-spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,µ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on L_p-spectral independence can be treated as special cases of our results and give examples—including strictly elliptic operators in Euclidean space and generators of semigroups that satisfy (generalized) Gaussian bounds—to indicate improvements.