Thermoregulatory behaviour can be viewed in the context of trade-offs between different behaviours that have associated costs and benefits. For example if a bird moves into the shade it may have to forego foraging. Males and females may simply be experiencing the cost and benefits differently due to different parental care roles. A female may be able to afford to stop foraging when it gets hot because she isn't trying to feed an entire brood of chicks, whereas a male may have to keep foraging. Alternatively a female may be unable to allow her body temperature to rise as she is having to shade eggs/nestlings from the sun.

There are also physical and physiological difference that may be important. There may differences in size, metabolic rate or perhaps elevated body temperatures may be detrimental during the development of eggs. 

Which species did you observe this in?

The species of interest is Cape Sugarbird which i'm working on.

One would expect them to respond equally to solar radiation.

I believe foraging behaviour may be behind sex differences. If males are food seekers they have to cope with heat unless food is available in shaded places. In any case, evaluating cooling costs should be determined in a specific experiment controlling temperature, air speed and humidity and taking into account body size.

You can contact to scientists zoologists Moscow University (Russia) on this theme  - they are on this site. I send one of articles of bird energy.

Could this simply be a case where sexual dimophism results in an additional cost to the male?  Is it possible that the male's plumage (particularly the tail) and overall larger size make it more difficult for them to physically "fit" into habitat that affords shade?

Yes, often the beauty of the male is for their rapid fertilization of females and for quick dying for the sake of posterity. But the beauty is still necessary and important.

Hi.

It may be interesting for you to read the attached paper, regarding to the cooling behaviour of some species of larks living in desert environments of the Middle East.

Best regards. 

There are high thermoregulatory costs to incubation. In hot environments, incubating birds seek ways to cool themselves, such as seeking shaded nest sites, or bellysoaking (observed in many shorebirds). If the species you are interested in has uniparental incubation by the females (or at least females are the only ones incubating during the hottest part of the day), then it would make sense that you would see females, but not males, engaging in behaviors aimed at cooling themselves. This is because females would be experiencing higher thermal stress while incubating eggs.