Thermal quantum critical systems, with partition functions expressed as conformal tensor networks, are revealed to exhibit universal entropy corrections on nonorientable manifolds. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy SK=lnk on the Klein bottle, where k relates to quantum dimensions of the primary fields in conformal field theory (CFT). Different from the celebrated Affleck-Ludwig boundary entropy lng (g reflects noninteger ground-state degeneracy), SK has no boundary dependence or surface energy terms accompanying it, and can be very conveniently extracted from thermal data. On the Möbius-strip manifold, we uncover an entropy SM=12(lng+lnk) in CFT, where 12lng is associated with the only open edge of the Möbius strip and 12lnk is associated with the nonorientable topology. As a useful application, we employ the universal entropy to accurately pinpoint the quantum phase transitions, even for those without local order parameters.