  In this paper we study a model for phase separation and damage in
thermoviscoelastic materials. The main novelty of the paper consists in the
fact that, in contrast with previous works in the literature (cf., e.g., [C.
Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard
systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011),
321--359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface
models describing phase separation and damage. European J. Appl. Math. 24
(2013), 179--211]), we encompass in the model thermal processes, nonlinearly
coupled with the damage, concentration and displacement evolutions. More in
particular, we prove the existence of "entropic weak solutions", resorting to a
solvability concept first introduced in [E. Feireisl: Mathematical theory of
compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53
(2007), 461--490] in the framework of Fourier-Navier-Stokes systems and then
recently employed in [E. Feireisl, H. Petzeltov\'a, E. Rocca: Existence of
solutions to a phase transition model with microscopic movements. Math. Methods
Appl. Sci. 32 (2009), 1345--1369], [E. Rocca, R. Rossi: "Entropic" solutions to
a thermodynamically consistent PDE system for phase transitions and damage.
SIAM J. Math. Anal., 47 (2015), 2519--2586] for the study of PDE systems for
phase transition and damage. Our global-in-time existence result is obtained by
passing to the limit in a carefully devised time-discretization scheme.
