The present invention relates to a flexible composite organic aerogel (1) comprising: a textile reinforcement (5), an organic aerogel (3) placed within said textile reinforcement (3), said organic aerogel (3) being based on a resin resulting at least in part from polyhydroxybenzene(s) R and formaldehyde(s) F, said organic aerogel (3) being a polymeric organic gel comprising at least one water-soluble cationic polyelectrolyte, or said organic aerogel (3) being a pyrolysate of said gel in the form of a porous carbon monolith comprising the product of the pyrolysis of said at least one water-soluble cationic polyelectrolyte P, said organic aerogel (3) exhibiting a specific thermal conductivity of between 10 and 40 mW·m−1·K−1 at atmospheric pressure.