The principal goal of this paper is to use attempts at reconciling the Swift long gamma-ray bursts (LGRBs) with the star formation history (SFH) to compare the predictions of Λ cold dark matter (ΛCDM) with those in the Rh = ct Universe. In the context of the former, we confirm that the latest Swift sample of GRBs reveals an increasing evolution in the GRB rate relative to the star formation rate (SFR) at high redshifts. The observed discrepancy between the GRB rate and the SFR may be eliminated by assuming a modest evolution parametrized as (1 + z)0.8 - perhaps indicating a cosmic evolution in metallicity. However, we find a higher metallicity cut of Z = 0.52 Z☉ than was seen in previous studies, which suggested that LGRBs occur preferentially in metal-poor environments, i.e. Z ̃ 0.1-0.3 Z☉. We use a simple power-law approximation to the high-z ( ≳ 3.8) SFH, i.e. RSF ∝ [(1 + z)/4.8]α, to examine how the high-z SFR may be impacted by a possible abundance evolution in the Swift GRB sample. For an expansion history consistent with ΛCDM, we find that the Swift redshift and luminosity distributions can be reproduced with reasonable accuracy if α =-2.41_{-2.09}^{+1.87}. For the Rh = ct Universe, the GRB rate is slightly different from that in ΛCDM, but also requires an extra evolutionary effect, with a metallicity cut of Z = 0.44 Z☉. Assuming that the SFR and GRB rate are related via an evolving metallicity, we find that the GRB data constrain the slope of the high-z SFR in Rh = ct to be α =-3.60_{-2.45}^{+2.45}. Both cosmologies fit the GRB/SFR data rather well. However, in a one-on-one comparison using the Akaike information criterion, the best-fitting Rh = ct model is statistically preferred over the best-fitting ΛCDM model with a relative probability of ̃70 per cent versus ̃30 per cent.