Single-walled carbon nanotube (SWCNT) aerogels produced by critical-point-drying of wet-gel precursors exhibit unique properties, such as high surface-area-to-volume and strength-to-weight ratios. They are free-standing, are binder-free, and can be scaled to thicknesses of more than 1mm. Here, we examine the electric double layer capacitive behavior of these materials using a common room temperature ionic liquid electrolyte, 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMI-TFSI). Electrochemical performance is assessed through galvanostatic cycling, cyclic voltammetry and impedance spectroscopy. Results indicate stable capacitive performance over 10,000 cycles as well as an impressive performance at high charge and discharge rates, due to accessible pore networks and enhanced electronic and ionic conductivities of SWCNT aerogels. These materials can find applications in mechanically compressible and flexible supercapacitor devices with high power requirements.