R: Optimize algorithm speed that estimates hourly temperature and relative humidity
The first thing to improve about your code/procedure is readability.
Not "just" for others trying to understand your code, but for your latter self should you want to maintain it.
I don't know the first thing about R. The first style guide a superficial web search turned up is from Hadley Wickham's Advanced R, characterised as short and sweet.

Things I think to harm readability:


(lack of) structuring source code by formatting
(lack of) documentation in the source code: What is "everything" about?
What is the problem to solve, the sequence of well-defined steps perceived to solve it?
abbreviated names without an illuminating comment: is "hr" for hour or  humidity, relative or something entirely else?
hx seems to be four hours before sunset - what does the x signify, and why 4 hours?
in-line arithmetic with magical constants: while I can guess that lat*(3.141592653589793/180.0) is radian(latitude in degrees), I'd rather not.
What is the significance of 10 in dayIndex + 10,
0.39 in tMax - 0.39*(tMax - tMinNext), why isn't the latter just 0.39*tMinNext + 0.61*tMax?
computing things that don't get used
The only things I see used (pasted to hrResult - see below) are hr and tSum;
some code struggles with HumHr/HumTm and "their Accum counterparts".


Once the code is readable, one can start pondering performance impact:
Is there anything straining the memory hierarchy?
Do costly actions get repeated instead of results reused?
Then, there is general advice like Use Vectorisation,
and (resource demand) pitfalls - using paste() to accumulate the result may be one.