Based on a fiber bundle model we substantially extend the phase-transition analogy of thermally activated breakdown of homogeneous materials. We show that the competition of breaking due to stress enhancement and due to thermal fluctuations leads to an astonishing complexity of the phase space of the system: varying the load and the temperature a phase boundary emerges, separating a Griffith-type regime of abrupt failure analogous to first-order phase transitions from disorder dominated fracture where a spanning cluster of cracks emerges. We demonstrate that the phase boundary is the Kertész line of the system along which thermally activated fracture appears as a continuous phase transition analogous to percolation. The Kertész line has technological relevance setting the boundary of safe operation for construction components under high thermal loads. © 2010 The American Physical Society.