We study the phenomena of decrease in lower critical solution temperature (LCST) with addition of kosmotropic (order-making) cosolvents in thermoresponsive polymer solutions. A combination of explicit solvent coarse-grained simulations and mean-field theory has been employed. The polymer-solvent LCST behavior in the theoretical models has been incorporated through the Kolomeisky-Widom solvophobic potential. Our results illustrate how the decrease in the LCST can be achieved by the reduction in the bulk solvent energy with the addition of cosolvent. It is shown that this effect of cosolvent is weaker with an increase in polymer hydrophilicity which can explain the absence of a LCST decrease in poly(N,N-diethylacrylamide), water, and methanol systems. The coarse-grained nature of the models indicates that a mean energetic representation of the system is sufficient to understand the phenomena of LCST decrease.