Thermal modifications of mesons and energy-energy correlators from real-time simulations of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:

We investigate thermal properties of a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> lattice gauge theory in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mn>1</e:mn> <e:mo>+</e:mo> <e:mn>1</e:mn> </e:math> dimensions through real-time simulations. We extract the spectral functions directly coupling to the pseudoscalar and scalar mesons, demonstrating the thermal modifications of these states with increasing temperatures. Introducing the notion of energy-flow operators, we quantify the temporal buildup of correlations in the energy flows across the lattice. We demonstrate that energy-energy correlators fail to factorize to products of energy flows, both in the vacuum and at nonzero temperature, indicating the presence of nontrivial correlations in the quantum states. Our results constitute a first real-time study of bound-state thermal broadening and finite temperature energy-flow correlations in a gauge theory, providing a benchmark for future studies of hadronic matter under extreme conditions.