What to do if at a final divorce hearing the judge is aware that the wife is currently living with another man but orders the husband to pay spousal support anyway?

Texas does not have alimony, but instead has spousal support laws which provided for support in limited circumstances.  The judge is required to review a list of factors that include the ability of the other spouse to work, their present skill set, any disabilities, and the division of the rest of the marital estate.  If a spouse can show, after a review of several factors, that they do not have the ability to support themselves, then the judge can award spousal support. A judge's decision will be evaluated by what is called an "abuse of discretion" standard. This means that as long as what the judge did is reasonable and there is sufficient evidence to support a need for spousal support, then his decision will not be overturned on appeal. 

In your case, the judge had to have reviewed the factors set out in the statute and the other considerations that have developed through case law.  If you do not agree with the judge's decision, you have a couple of different options.  The first is to file a motion for reconsideration or new trial if it has been a short time since his ruling.  The judge can take into consideration that she is living with someone else, but he is allowed to limit the weight given to the cohabitation because the person is not legally obligated to support her and could require her to leave at any time.  If you have evidence that the relationship is going to be more permanent, your argument against spousal support would be stronger.  Your second option flows from the first option--- if the judge will not reconsider/overturn his decision, then ask the judge to place limitations on his decision.  For example, if she continue to live rent free for "x" number of months, that you get a credit on your spousal support.  You can also ask that a provision be included in the final order that if she remarries (legal or common law) that you be discharged from the spousal obligation.  Your

Your third option is to file a notice of appeal.  When an issued is evaluated by an "abuse of discretion" standard, it tends to be hard to get a trial court decision overturned on appeal---- but it does happen and is not impossible.  If you do not think the judge reviewed the factors appropriately or there was not evidence that she did not have the ability to support herself, then you could file a notice of appeal and ask a higher court to review what the trial court did.