Charge selective asymmetric nanochannels are used for a variety of applications, such as nanofluidic sensing devices and energy conversion applications. In this paper, we numerically investigate the influence of an applied temperature difference over tapered nanochannels on the resulting charge transport and flow behavior. Using a temperature-dependent formulation of the coupled Poisson-Nernst-Planck and Navier-Stokes equations, various nanochannel geometries are investigated. Temperature has a large influence on the total ion transport, as the diffusivity of ions and viscosity of the solution are strongly affected by temperature. We find that the selectivity of the nanochannels is enhanced with increasing asymmetry ratios, while the total current is reduced at higher asymmetry cases. Most interestingly, we find that applying a temperature gradient along the electric field and along the asymmetry direction of the nanochannel enhances the selectivity of the tapered channels even further, while a temperature gradient countering the electric field reduces the selectivity of the nanochannel. Current rectification is enhanced in asymmetric nanochannels if a temperature gradient is applied, independent of the direction of the temperature difference. However, the degree of rectification is dependent on the direction of the temperature gradient with respect to the channel geometry and the electric field direction. The enhanced selectivity of nanochannels due to applied temperature gradients could result in more efficient operation in energy harvesting or desalination applications, motivating experimental investigations.