I like things with menthol, and at the moment I'm working my way through to the end of a tub of OSP Grapefruit anf Menthol soap. When that's gone, I'll have to chose something else, as OSP is no longer making.
For splashes, I like Floid Vigoroso, which has a menthol constituent. The other stuff that's refreshing on a hot day is witch hazel; either Humphrey's or one of the Thayers lightly-scented ones.
I would definitely recommend cold water on a hot day, not hot water. In fact I'd recommend cold water on any day, but certainly on hot days. It will help
My coldest set-up is Proraso Green pre-shave, P&B Denali or ESC Fiat Ach Snog for the soap or cream followed by Floïd Vigaroso ASL and Proraso Green ASB. There's much 'colder' products out there but this works for me. Also a nice set-up when full of cold or starting the day after a poor night's sleep.
No preshave. OSP Grapefruit and Menthol, or Phoenix and Beau Denali soap, after shave? Depending on the cool ‘hit' that I want, any from Proraso Menthol and Eucalyptus, Flöid Mentally Vigorous or Alcoloda Glacial. But if I want something approaching intracranial cryogenics, Fine Snake Bite.
Hot days? Yes I think there was one of those in 1976 - my ne plus ultra for menthol - cold water - P&B Denali or MT Mint Ice Menthol - Alcolada Glacial - its got a penguin on the label - just in case you don't understand what is about to happen - then - Vifrex gel. None more frosty. I.