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Observational evidence for the accelerated expansion of the universe requires dark energy for its explanation if general relativity is an accurate model of gravity. However, dark energy is a mysterious quantity and we do not know much about its nature so understanding dark energy is an exciting scientific challenge. Cosmological dark energy models are fairly well tested in the low and high redshift parts of the universe. The highest of the low redshift, $z\sim2.3$, region is probed by baryon acoustic oscillation (BAO) measurements and the only high redshift probe is the cosmic microwave background anisotropy which probes the $z\sim1100$ part of redshift space. In the intermediate redshift range $2.3 < z < 1100$ there are only a handful of observational probes and cosmological models are poorly tested in this region. In this thesis we constrain three pairs of general relativistic cosmological dark energy models using observational data which reach beyond the current BAO limit. We use quasar X-ray and UV flux measurements, the current version of these data span $0.009 \leq z \leq 7.5413$. We have discovered that most of these data cannot be standardized using the proposed method. However, the lower redshift part, $z \lesssim 1.5-1.7$, of these data are standardizable and can be used to derive lower-$z$ cosmological constraints. Another data set we use are gamma-ray burst measurements which span $0.3399 \leq z \leq 8.2$. Cosmological constraints derived from these data are significantly weaker than, but consistent with, those obtained from better-established cosmological probes. We also study and standardize 78 reverberation-measured Mg II time-lag quasars in the redshift range $0.0033 \leq z \leq 1.89$ by using their radius-luminosity relation. We also study 118 reverberation-measured H$β$ time-lag quasars which span $0.0023 \leq z \leq 0.89$.