  Partition functions of quantum critical systems, expressed as conformal
thermal tensor networks, are defined on various manifolds which can give rise
to universal entropy corrections. Through high-precision tensor network
simulations of several quantum chains, we identify the universal entropy
$S_{\mathcal{K}} = \ln{k}$ 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 $\ln{g}$ ($g$ reflects
non-integer groundstate degeneracy), $S_{\mathcal{K}}$ has \textit{no} boundary
dependence or surface energy terms accompanied, and can be very conveniently
extracted from thermal data. On the M\"obius-strip manifold, we uncover an
entropy $S_{\mathcal{M}} = \frac{1}{2} (\ln{g} + \ln{k})$ in CFT, where
$\frac{1}{2} \ln{g}$ is associated with the only open edge of the M\"obius
strip, and $\frac{1}{2} \ln{k}$ with the non-orientable topology. We employ
$S_{\mathcal{K}}$ to accurately pinpoint the quantum phase transitions, even
for those without local order parameters.
