Q: What is the rule regarding a man who doesn’t perform salaat with a topi (kufya, headcovering), and what is the shari rule regarding the covering of the head during salaat?
A: Although it was the practise of Rasulullah sallalaha alaihi wasallam and the sahaaba to cover their heads at all times special emphasis was given to covering the head when performing salaah.
Imam Bukhari in his Saheeh mentions the statements of Hasan Basri that the Sahaaba kiraam used to make sajdah upon the turban and the topee. Ibn Asim mentions that he found Nabi (sallalahu alaihi wasallam) and the Sahaaba kiram performing salaah wearing the baraanees (a type of headcovering).
This shows us performing salaah with the head covered was the practice of our beloved Nabi (sallalahu alaihi wasallam) and our pious predecessors. Hence it is highly encouraged to do so. Failing to do so would be tantamount to compromising the sunnat of Nabi (sallalahu alaihi wasallam).