the question “does ozai actually love azula?” is really interesting because there really isn’t a right answer. it’s clear that ozai likes azula, much in the way that i like the pen i keep in my bag, but will also not hesitate to discard it once it runs out of ink. and if you consider love to be something that is unconditional (or at the very least, far less conditional than ozai’s standards for his children), then the obvious answer is no, he doesn’t. but that doesn’t mean he doesn’t feel a strong affection for azula, a pride for the way in which her successes reflect well on his nation, his crown, his ego (all one & the same to him anyway).

does he think he loves her? i mean, yeah. sure. he probably also thinks of himself as loving the people of the fire nation—after all, everything he does is in the best interest of his country.

he learned the values taught to him by his father, whose “love” for ozai is clearly also very tenuous. iroh was clearly the favorite, and ozai the the discarded one. so ozai simply expresses the behaviors that he’s learned. he superficially replicates azulon’s relationship with iroh in his relationship with azula while treating zuko like a punching bag, offloading his insecurities onto someone he can only seem to see as the personification of all of his own father issues and neuroses.

obviously ozai sees his wife and children as his property – that’s not even a particularly unique viewpoint for any man (in a patriarchal society) to hold regardless of their social status – but as reflections of himself, anything they do that reflects poorly on him requires extreme consequences to mitigate, seeing as he is in a position of extreme power that he is also well aware wasn’t entirely “deserved.” both ozai and azula banish people when they feel threatened, because to the crown, a threat to the state (ursa and zuko both proving that they are capable of treason) is a threat to their ego (azula's extreme paranoia during her breakdown) and vice versa.

but i find it especially ironic that the only person it's clear ozai genuinely has any kind of real appreciation for is zhao, because zhao's defining qualities of hotheaded impulsiveness and ego at the expense of logic are the exact qualities that ozai would find utterly repulsive in his children – despite, of course, possessing these qualities himself. ozai punishes zuko's impulsive behavior (a quality he himself was likely punished for by azulon) while appreciating azula's tact, patience, rationality, strategy, and ability to sublimate her own ego for the sake of loyalty to her father/nation (all traits that were also admired in iroh, mind you).

when ozai sees his own worst qualities mirrored in his son, he banishes him, wants him dead. he all but writes off azula after he learns that she lied to him, and seems disgusted by her after her outburst. but seeing this egomaniacal, violent drive reflected in zhao is somehow different, and it leads to his constant promotion. in fact, he only thinks to utilize azula to capture the avatar/neutralize zuko once zhao has finally perished at the hand of his own hubris, even though azula is the far more competent, efficient asset, not only in intellect and skill, but also because her loyalty far exceeds her ego.

the irony of it all, that ozai rewards zhao for relecting the worst parts of himself whereas his children have absolutely no leeway, is likely due to the paternal instinct to view his children as his legacy, his property, his burdens, potential threats to his throne/life (the way he was to his father), whereas zhao is just a buddy, not blood.

but back to the original question: does ozai love azula? what is love, anyway? it's pretty clear ozai would kill any of his family members without hesitation if he thinks it would allow him to seize/maintain power. but azula never gives her father a reason to believe that her goals are not aligned with, defined by, his except for when she brings zuko home and secures his position as heir by lying to his face, so she's obviously his favorite family member. but she's his child, and she's a girl, so as much pride and affection as he feels for her, it's clear he also views her as his property, as a weapon. then again, i doubt ozai thinks of himself as some mustache-twirling child-manipulator. i doubt he considers himself evil; he's a powerful man who does what needs to be done. and all told, he's a pretty normal byproduct of his environment.

so no, I don’t think ozai truly loves azula. but he thinks he does. after all, he treats her far better than his father ever treated him.