We study the effect of a stationary heat current Q on superfluid He-4 in a homogeneous metastable state near T(lambda)(Q). On the basis of a renormalization-group calculation we predict a sizable enhancement of the specific heat C(P)(Q) and a weak depression of the superfluid density rho(s)(Q) up to a critical heat current Q(c)(T) where C(P) and rho(s) exhibit cusplike anomalies.