  Universal thermal data in conformal field theory (CFT) offer a valuable means
for characterizing and classifying criticality. With improved tensor network
techniques, we investigate the universal thermodynamics on a nonorientable
minimal surface, the crosscapped disk (or real projective plane,
$\mathbb{RP}^2$). Through a cut-and-sew process, $\mathbb{RP}^2$ is
topologically equivalent to a cylinder with rainbow and crosscap boundaries. We
uncover that the crosscap contributes a fractional topological term
$\frac{1}{2} \ln{k}$ related to nonorientable genus, with $k$ a universal
constant in two-dimensional CFT, while the rainbow boundary gives rise to a
geometric term $\frac{c}{4} \ln{\beta}$, with $\beta$ the manifold size and $c$
the central charge. We have also obtained analytically the logarithmic rainbow
term by CFT calculations, and discuss its connection to the renowned
Cardy-Peschel conical singularity.
