The thermal conductivity of gas-permeated single-walled carbon nanotube (SWCNT) aerogel (8 kg m-3 density, 0.0061 volume fraction) is measured experimentally and modeled using mesoscale and atomistic simulations. Despite the high thermal conductivity of isolated SWCNTs, the thermal conductivity of the evacuated aerogel is 0.025 +/- 0.010 W m-1 K-1 at a temperature of 300 K. This very low value is a result of the high porosity and the low interface thermal conductance at the tubetube junctions (estimated as 12 pW K-1). Thermal conductivity measurements and analysis of the gas-permeated aerogel (H2, He, Ne, N2, and Ar) show that gas molecules transport energy over length scales hundreds of times larger than the diameters of the pores in the aerogel. It is hypothesized that inefficient energy exchange between gas molecules and SWCNTs gives the permeating molecules a memory of their prior collisions. Low gas-SWCNT accommodation coefficients predicted by molecular dynamics simulations support this hypothesis. Amplified energy transport length scales resulting from low gas accommodation are a general feature of CNT-based nanoporous materials.