In this paper, we demonstrate experimentally that field-effect transistors (FETs) with nanoconstricted graphene monolayer channels have a subthreshold swing (SS) below 60 mV/dec, which is slightly dependent on temperature. Two shapes of nanoconstricted graphene monolayers are considered: (i) a bow-tie shape, representative for a symmetric channel, and (ii) a trapezoidal shape, which illustrates an asymmetric channel. While both types of nonuniform channels are opening a bandgap in graphene, thus showing an on/off ratio of 105, the SS in the graphene bow-tie channel is below 60 mV/dec in the temperature range 25 oC-44 oC.