You don't say what your originating city is. That aside, the stopover rules are specific to each fare, and are not general airline rules. Generally speaking, the more expensive the fare, the more flexible the stopover rules are. They go from "no stopovers permitted" to "stopovers permitted at $x surcharge" to "free unlimited stopovers", depending on the fare.

I picked PHL as an originating city just for grins, going to HNL, 4/10-4/20 (so, no advanced purchase).

The cheapest AA fare says " 2 STOPOVERS PERMITTED ON THE PRICING UNIT - 1 IN EACH DIRECTION IN DFW/CHI/LAX/SFO AT USD 55.81 EACH. NO STOPOVER OCCURS IF PASSENGER TAKES NEXT AVAILABLE FLIGHT WITHIN 4 HOURS." So in that case, it would not work for you, as a stopover is defined as being anything over 4 hours, and is permitted only in certain cities.

The cheapest US fare says "NO STOPOVERS PERMITTED ON THE PRICING UNIT. NO STOPOVER OCCURS IF PASSENGER TAKES NEXT AVAILABLE FLIGHT WITHIN 18 HOURS." So, in that case, you could spend the night in PHX, as long as the flight the next morning was a) the next flight, and b) within 18 hours of your arrival.

Note that you have to read the fare rules carefully: in the US case cited above, the rule is very clear: NO STOPOVERS, but then it goes on to define a "stopover" in a way that what you want to do is not considered a stopover. This is important, as when you're shopping around, you don't want to tell an airline or travel agent that you "need a stopover in PHX" as an inexperienced agent might simply gloss over any fare that says NO STOPOVERS.

Looking forward a month so that deep discount fares are available, the prices are a lot lower, but the stopover rules are the same. I used ExpertFlyer to do the research.