We present the results of a first study of the equation of state in finite-temperature QCD with two flavors of Wilson-type quarks. The simulations are made on lattices with a temporal size Nt=4 and 6, using a renormalization-group-improved action for the gluon sector and a mean-field-improved clover action for the quark sector. The lines of constant physics corresponding to fixed values of the ratio mPS/mV of the pseudoscalar to vector meson masses at zero temperature are determined, and the beta functions which describe the renormalization-group flow along these lines are calculated. Using these results, the energy density and the pressure are calculated as functions of temperature along the lines of constant physics in the range mPS/mV=0.65–0.95. The quark mass dependence in the equation of state is found to be small for mPS/mV≲0.8. A comparison of the results for Nt=4 and Nt=6 lattices shows a significant scaling violation present in the Nt=4 results. At high temperatures the results for Nt=6 are quite close to the continuum Stefan-Boltzmann limit, suggesting the possibility of a precise continuum extrapolation of thermodynamic quantities from simulations at Nt≳6.