Which star models are thermodynamically acceptable in Einstein's gravity?

Hundreds of exact solutions to Einstein's equations describe perfect-fluid stars, but most violate physical common sense. This work adds thermodynamic consistency — entropy must be finite, positive, and match the Tolman temperature relation — as a new screening criterion, testing it on the classic Tolman IV solution, which passes. The approach offers a principled way to discard mathematical curiosities and identify which relativistic stellar models are genuinely physical.