the recovery is what, 6 or 7 years old?
The housing market bottomed at the end of 2011 and the growth started around 2012. So it is less than 6 or 7 
increased interest rates will be small and not significantly reduce the number of qualified buyers
In the past when I used the average monthly mortgage and the impact of that on the interest rate. That is, suppose LGIH has an average selling price of $200 K a @3.5% mortgage rate monthly payment could be around $890 vs $955 for 4% vs $1013 for 4.5%.
So the $125 increase is basically 15% higher monthly mortgage cost. Since LGIH caters to starters, people who are buying their first house, this is a segment that has very little disposable income and often with young families, a 15% monthly mortgage payment could evaporate demand quickly. My rule of thumb is demand could be impacted by 10% to 20%.
Typically, increasing rates also mean economy improving, thus higher employment and increasing wages. But we are already at record employment (please I am not interested in any political debate on this) and I haven’t seen any wage growth (at least for me certainly 
). The last few years we have benefited from low gas price and that could turn out to be a headwind.
So the impact on demand could be much severe and I would also suggest you to check out New home closing during “taper tantrum” period, I believe we have seen some wild bounces at least for the companies I followed at that time.
Why Does LGIH Have a Low P/E?
Typically home builders trade around 9~12 PE. So I am not seeing a significant discount. The last few years are outlier as the homebuilders were bouncing back from closings and many have carried significant deferred tax asset’s and skewing their EPS, etc.
A word on debt
Your concern is misplaced. I took a quick look at the 10Q and I see the debt is mainly revolver and a $77M 4.5% convertible notes. The company relying on revolver is a concern, i.e lack of permanent debt financing should be the concern.
Since LGIH builds spec’s (speculative building without a firm order), they need to maintain an inventory of houses to sell, that means typically 6 months of sales will be in inventory, this includes completed houses and houses that are under construction, etc and there are seasonable variations like spring sales would require higher inventory etc. But you should basically expect 6 months of inventory and land inventory. In an earlier post I mentioned LGIH maintains 7+ years of lot inventory. In summary I don’t think LGIH debt is high but I do have a concern that they are not having permanent financing and relying on revolver. It may be beneficial to lock in low rates. Perhaps they are not able to issue debt at reasonable cost hence they decided to stick with revolver (which are often lower rates but gets pulled out at the wrong time).