Reionization leads to large spatial fluctuations in the intergalactic temperature that can persist well after its completion. We study the imprints of such fluctuations on the z ̃ 5 Ly α forest flux power spectrum using a set of radiation-hydrodynamic simulations that model different reionization scenarios. We find that large-scale coherent temperature fluctuations bring {̃}20-60{{ per cent}} extra power at k ̃ 0.002 s km-1, with the largest enhancements in the models where reionization is extended or ends the latest. On smaller scales (k ≳ 0.1 s km-1), we find that temperature fluctuations suppress power by {≲}10{{ per cent}}. We find that the shape of the power spectrum is mostly sensitive to the reionization mid-point rather than temperature fluctuations from reionization's patchiness. However, for all of our models with reionization mid-points of z ≤ 8 (z ≤ 12), the shape differences are {≲}20{{ per cent}} ({≲}40{{ per cent}}) because of a surprisingly well-matched cancellation between thermal broadening and pressure smoothing that occurs for realistic thermal histories. We also consider fluctuations in the ultraviolet background, finding their impact on the power spectrum to be much smaller than temperature fluctuations at k ≳ 0.01 s km-1. Furthermore, we compare our models to power spectrum measurements, finding that none of our models with reionization mid-points of z < 8 is strongly preferred over another and that all of our models with mid-points of z ≥ 8 are excluded at 2.5σ. Future measurements may be able to distinguish between viable reionization models if they can be performed at lower k or, alternatively, if the error bars on the high-k power can be reduced by a factor of 1.5.
