The stability of fault slip is sensitive to the way in which frictional strength responds to changes in slip rate and in particular to the effective velocity dependence of steady state friction Δμss/Δ ln V. This quantity can vary substantially with displacement, temperature and slip rate. To investigate the physical basis for this behavior and the possible influence of shear heating, we slid initially bare granite surfaces in unconfined rotary shear to displacements of hundreds of millimeters at normal stresses, σn of 10 and 25 MPa and at room temperature. We imposed step changes in slip rate within the range 10−2 to 103.5 μm/s and also monitored frictional heating with thermistors embedded in the granite. The transient response of μ to slip rate steps was fit to a rate‐ and state‐dependent friction law using two state variables to estimate the values of several parameters in the constitutive law. The first 20 mm of slip shows rising friction and falling Δμss/Δ ln V; further slip shows roughly constant friction, Δμss/Δ ln V and parameter values, suggesting that a steady state condition is reached on the fault surface. At V ≤ 10 μm/s, Δμss/Δ ln V = −0.004 ± 0.001. At higher rates the response is sensitive to normal stress: At σn = 25 MPa granite shows a transition to effective velocity strengthening (Δμss/Δ ln V = 0.008 ± 0.004) at the highest slip rates tested. At 10 MPa granite shows a less dramatic change to Δμss/Δ ln V ≈ 0 at the highest rates. The maximum temperature measured in the granite is ∼60°C at 25 MPa and 103.5 μm/s. Temperatures are in general agreement with a numerical model of heat conduction which assumes spatially homogeneous frictional heating over the sliding surface. The simplest interpretation of our measurements of Δμss/Δ ln V is that the granite is inherently velocity weakening (∂μss/∂ ln V < 0) and temperature strengthening (∂μss/∂T−1 < 0) at all velocities. At high slip rates the response of μ to changes in temperature from shear heating may outweigh the response to changing velocity, such that the net effect Δμss/Δ ln V > 0 mimics velocity strengthening. These results have implications for the frictional behavior of faults during earthquakes. High slip rates may cause a switch to effective velocity strengthening which could limit peak coseismic slip rate and stress drop. For fluid‐saturated faults, strengthening by this mechanism may be partly or fully offset by weakening due to thermal pressurization of a poorly drained pore fluid.