Running Vacuum Cosmology in f(R,T) Gravity: Observational Constraints and Thermodynamic Analysis
Despite the remarkable observational success of the $\Lambda$CDM model, the physical origin of dark energy and the cosmological constant remain unresolved, motivating the exploration of dynamical vacuum scenarios within modified theories of gravity. Inspired by the quantum field theoretical description of running vacuum energy, we investigate a running vacuum cosmology in the framework of linear $f(R,T)=R+\lambda T$ gravity. Unlike recent studies that establish an effective correspondence between running vacuum energy and modified gravity, our approach directly incorporates a quantum field theory motivated running vacuum scenario into the $f(R,T)$ framework and examines its cosmological implications. Exact analytical solutions of the modified field equations are obtained in a spatially flat Friedmann--Lema\^itre--Robertson--Walker spacetime, yielding an analytical expression for the Hubble parameter. The free model parameters are constrained using the latest Pantheon+SH0ES compilation, DESI baryon acoustic oscillation measurements, and their joint dataset through Markov Chain Monte Carlo analysis. The observationally constrained model successfully reproduces the observed expansion history, remains consistent with independent Cosmic Chronometer measurements, predicts an age of the universe within current observational bounds, and yields an effective equation of state compatible with current observations. Furthermore, the statefinder and $Om$ diagnostics indicate close agreement with the $\Lambda$CDM scenario at late times, while the generalized second law of thermodynamics remains valid throughout cosmic evolution. These results demonstrate that the proposed framework provides an observationally viable and thermodynamically consistent realization of running vacuum cosmology in $f(R,T)$ gravity, offering a theoretically motivated alternative for describing the present accelerated universe.