  We reveal an intriguing anomaly in the temperature dependence of the specific
heat of a one-dimensional Bose gas. The observed peak holds for arbitrary
interaction and remembers a superfluid-to-normal phase transition in higher
dimensions, but phase transitions are not allowed in one dimension. The
presence of the anomaly signals a region of unpopulated states which behaves as
an energy gap and is located below the hole branch in the excitation spectrum.
The anomaly temperature is found to be of the same order of the energy of the
maximum of the hole branch. We rely on the Bethe Ansatz to obtain the specific
heat exactly and provide interpretations of the analytically tractable limits.
The dynamic structure factor is computed with the Path Integral Monte Carlo
method for the first time. We notice that at temperatures similar to the
anomaly threshold, the energy of the thermal fluctuations become comparable
with the maximal hole energy, leading to a qualitative change in the structure
of excitations. This excitation pattern experiences the breakdown of the
quasi-particle description for any value of the interaction strength at the
anomaly, similarly to any superfluid phase transition at the critical
temperature. We provide indications for future observations and how the hole
anomaly can be employed for in-situ thermometry, identifying different
collisional regimes and understanding other anomalies in atomic, solid-state,
electronic, spin-chain and ladder systems.
