  Cosmological models with variable and modified equations of state for dark
energy are confronted with observational data, including Type Ia supernovae,
Hubble parameter data $H(z)$ from different sources, and observational
manifestations of cosmic microwave background radiation (CMB). We consider
scenarios generalizing the $\Lambda$CDM, $w$CDM, and
Chevallier--Polarski--Linder (CPL) models with nonzero curvature and compare
their predictions. The most successful model with the dark energy equation of
state $w= w_0+ w_1(1-a^2)/2$ was studied in detail. These models are
interesting in possibly alleviating the Hubble constant $H_0$ tension, but they
achieved a modest success in this direction with the considered observational
data.
