So, could long-term changes in the Sun's energy output have caused the change in Earth's climate measured over the last 35 years?
By and large, the space-based observations of the last 35 years have not recorded substantial changes in energy output from the Sun. Nonetheless, scientists include all the influences they can (including solar changes) when studying changes in climate. These estimates suggest that a small decrease in solar irradiance over the last 35 years would have caused a slight cooling of the climate over this time period – but only in the absence of other influences on Earth's climate.
The physics of the situation also doesn't back up the idea that changes in the Sun are a large force behind current climate change. The Sun's irradiance has its greatest effect on Earth's upper atmosphere, while the lower atmosphere insulates Earth from the increased heat. If the Sun were driving Earth's warming, one would expect to see that upper atmosphere getting increasingly hot. Instead, measurements show the lower atmosphere is getting hotter, while the upper atmosphere is getting cooler. Instead, this matches the fingerprint of changes driven by increases in carbon dioxide much more closely.